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Ch 1Setsसमुच्चयSets

What a set is, roster and set-builder form, empty, finite, infinite and equal sets, subsets and intervals, universal set, Venn diagrams, union, intersection, difference and complement — with every NCERT exercise solved.समुच्चय क्या है, रोस्टर व समुच्चय-निर्माण रूप, रिक्त, परिमित, अपरिमित व समान समुच्चय, उपसमुच्चय और अंतराल, सार्वत्रिक समुच्चय, वेन आरेख, सम्मिलन, सर्वनिष्ठ, अंतर और पूरक — NCERT के सभी प्रश्नों के हल के साथ।Set kya hai, roster aur set-builder form, empty, finite, infinite aur equal sets, subsets aur intervals, universal set, Venn diagrams, union, intersection, difference aur complement — NCERT ke saare questions ke hal ke saath.

Didn’t understand a paragraph? Tap it (or its अ button) → हिंदी / Hinglish / Simple English.कोई पैराग्राफ़ समझ नहीं आया? उस पर (या अ बटन पर) टैप करें → English / Hinglish।Koi paragraph samajh nahi aaya? Us par (ya अ button par) tap kijiye → हिंदी / English.

👆 Words with a open a diagram or graph — tap any label to read what it means.👆 वाले शब्द पर टैप करें — चित्र या ग्राफ़ खुलेगा (हर लेबल पर टैप = उसका अर्थ)।👆 wale word par tap kijiye — diagram ya graph khulega (har label par tap = uska matlab).
📘 NCERT chapter (English PDF)📗 NCERT हिंदी PDF∑ Formulas✎ NCERT solutions🎯 Quiz

Diagrams in this chapterइस अध्याय के चित्रIs chapter ke diagrams

Venn diagrams: union, intersection, difference, complementवेन आरेख: सम्मिलन, सर्वनिष्ठ, अंतर, पूरकVenn diagrams: union, intersection, difference, complementSections 1.9, 1.10Intervals on the real number lineवास्तविक संख्या रेखा पर अंतरालReal number line par intervalsSection 1.6.1Subset (A ⊂ B) and disjoint setsउपसमुच्चय (A ⊂ B) और असंयुक्त समुच्चयSubset (A ⊂ B) aur disjoint setsSections 1.6, 1.8
In this chapter (NCERT sections)इस अध्याय में (NCERT खंड)Is chapter mein (NCERT sections)
  1. 1.2Sets and their Representationsसमुच्चय और उनका निरूपणSets aur unka representation
  2. 1.3The Empty Setरिक्त समुच्चयEmpty Set
  3. 1.4Finite and Infinite Setsपरिमित और अपरिमित समुच्चयFinite aur Infinite Sets
  4. 1.5Equal Setsसमान समुच्चयEqual Sets
  5. 1.6Subsetsउपसमुच्चयSubsets
  6. 1.7Universal Setसार्वत्रिक समुच्चयUniversal Set
  7. 1.8Venn Diagramsवेन आरेखVenn Diagrams
  8. 1.9Operations on Setsसमुच्चयों पर संक्रियाएँSets par operations
  9. 1.10Complement of a Setसमुच्चय का पूरकSet ka Complement
  10. ★Yaad rakho — key formulasयाद रखो — मुख्य सूत्रYaad rakho — key formulas
  11. ⚠Common mistakesआम गलतियाँCommon mistakes
  12. ✎NCERT exercises — solutionsNCERT प्रश्नावली — हलNCERT exercises — solutions
  13. 🎯Quizक्विज़Quiz

1.2Sets and their Representationsसमुच्चय और उनका निरूपणSets aur unka representation

A set is a well-defined collection of objects — "well-defined" means that for any object we can say for sure whether it belongs or not. "Odd natural numbers less than 10" is a set: {1, 3, 5, 7, 9}. "The five most renowned mathematicians of the world" is not a set, because "most renowned" differs from person to person.समुच्चय वस्तुओं का सुपरिभाषित संग्रह है — "सुपरिभाषित" का अर्थ है कि किसी भी वस्तु के बारे में निश्चित रूप से कहा जा सके कि वह संग्रह में है या नहीं। "10 से कम विषम प्राकृत संख्याएँ" समुच्चय है: {1, 3, 5, 7, 9}। "विश्व के पाँच सबसे प्रसिद्ध गणितज्ञ" समुच्चय नहीं है, क्योंकि "सबसे प्रसिद्ध" हर व्यक्ति के लिए अलग है।Set objects ka well-defined collection hai — yaani kisi bhi object ke baare mein pakka keh sakein ki wo collection mein hai ya nahi. "10 se kam odd natural numbers" set hai: {1, 3, 5, 7, 9}. "Duniya ke 5 sabse famous mathematicians" set nahi hai, kyunki "sabse famous" har insaan ke liye alag hai.

Objects of a set are its elements. We write a \in A for "a belongs to A" and b \notin A for "b does not belong to A". Standard sets: \mathbf{N} natural numbers, \mathbf{Z} integers, \mathbf{Q} rational numbers, \mathbf{R} real numbers, \mathbf{Z}^+ positive integers.समुच्चय की वस्तुएँ उसके अवयव कहलाती हैं। "a, A का अवयव है" को a \in A और "b, A का अवयव नहीं है" को b \notin A लिखते हैं। मानक समुच्चय: \mathbf{N} प्राकृत संख्याएँ, \mathbf{Z} पूर्णांक, \mathbf{Q} परिमेय संख्याएँ, \mathbf{R} वास्तविक संख्याएँ, \mathbf{Z}^+ धन पूर्णांक।Set ki cheezein uske elements kehlati hain. "a, A ka element hai" ko a \in A aur "b, A ka element nahi hai" ko b \notin A likhte hain. Standard sets: \mathbf{N} natural numbers, \mathbf{Z} integers, \mathbf{Q} rational, \mathbf{R} real numbers, \mathbf{Z}^+ positive integers.

Two ways to write a set(i) Roster form — list all elements inside braces: \{2, 4, 6, 8\}. Order does not matter and an element is not repeated: letters of SCHOOL = {S, C, H, O, L}. (ii) Set-builder form — state the common property: \{x : x \in \mathbf{N},\ 3 < x < 10\}. ":" is read "such that".समुच्चय लिखने की दो विधियाँ(i) रोस्टर रूप — सभी अवयव कोष्ठक { } में: \{2, 4, 6, 8\}। क्रम मायने नहीं रखता और अवयव दोहराया नहीं जाता: SCHOOL के अक्षर = {S, C, H, O, L}। (ii) समुच्चय-निर्माण रूप — अवयवों का सर्वनिष्ठ गुण: \{x : x \in \mathbf{N},\ 3 < x < 10\}। ":" को "इस प्रकार है कि" पढ़ते हैं।Set likhne ke do tareeke(i) Roster form — saare elements braces { } mein: \{2, 4, 6, 8\}. Order matter nahi karta aur element repeat nahi hota: SCHOOL ke letters = {S, C, H, O, L}. (ii) Set-builder form — common property likhna: \{x : x \in \mathbf{N},\ 3 < x < 10\}. ":" ko "such that" padhte hain.
Example 3Write the solution set of x^2 + x - 2 = 0 in roster form.उदाहरण 3समीकरण x^2 + x - 2 = 0 के हल समुच्चय को रोस्टर रूप में लिखिए।Example 3x^2 + x - 2 = 0 ka solution set roster form mein likhiye.
Solution — step by step (1)हल — चरण दर चरण (1)Hal — step by step (1)
  1. x^2 + x - 2 = (x - 1)(x + 2) = 0 \Rightarrow x = 1,\ -2

✔ \{1, -2\}

Example 5Write \{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}\} in set-builder form.उदाहरण 5\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}\} को समुच्चय-निर्माण रूप में लिखिए।Example 5\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}\} ko set-builder form mein likhiye.
Solution — step by step (1)हल — चरण दर चरण (1)Hal — step by step (1)
  1. Each element is \frac{n}{n+1} with n = 1 to 6.हर अवयव \frac{n}{n+1} है, n = 1 से 6।Har element \frac{n}{n+1} hai, n = 1 se 6.

✔ \{x : x = \frac{n}{n+1},\ n \in \mathbf{N},\ 1 \le n \le 6\}

1.3The Empty Setरिक्त समुच्चयEmpty Set

A set with no element is the empty (null, void) set, written \phi or \{\ \}. Example: \{x : 1 < x < 2,\ x \in \mathbf{N}\} = \phi — no natural number lies between 1 and 2.जिस समुच्चय में कोई अवयव नहीं होता वह रिक्त (शून्य) समुच्चय है: \phi या \{\ \}। उदाहरण: \{x : 1 < x < 2,\ x \in \mathbf{N}\} = \phi — 1 और 2 के बीच कोई प्राकृत संख्या नहीं।Jis set mein koi element nahi hota wo empty (null) set hai: \phi ya \{\ \}. Example: \{x : 1 < x < 2,\ x \in \mathbf{N}\} = \phi — 1 aur 2 ke beech koi natural number nahi.

1.4Finite and Infinite Setsपरिमित और अपरिमित समुच्चयFinite aur Infinite Sets

A set is finite if it is empty or its elements can be counted to a definite end; n(A) = number of elements. Otherwise it is infinite. Days of the week: finite (n = 7). \mathbf{N}, lines through a point, points on a line: infinite.यदि समुच्चय रिक्त हो या उसके अवयवों की गिनती निश्चित संख्या पर समाप्त हो जाए, तो वह परिमित है; n(A) = अवयवों की संख्या। अन्यथा वह अपरिमित है। सप्ताह के दिन: परिमित (n = 7)। \mathbf{N}, एक बिंदु से जाने वाली रेखाएँ, रेखा के बिंदु: अपरिमित।Agar set empty ho ya uske elements ki ginti pakki sankhya par khatam ho, to wo finite hai; n(A) = elements ki ginti. Warna infinite. Week ke din: finite (n = 7). \mathbf{N}, ek point se jaane wali lines, line ke points: infinite.

1.5Equal Setsसमान समुच्चयEqual Sets

A and B are equal (A = B) if they have exactly the same elements. Repetition and order do not matter: letters of FOLLOW and of WOLF give the same set {F, O, L, W}.A और B समान (A = B) हैं यदि उनमें बिल्कुल वही अवयव हों। दोहराव और क्रम से फ़र्क नहीं पड़ता: FOLLOW और WOLF के अक्षरों से एक ही समुच्चय {F, O, L, W}।A aur B equal (A = B) hain agar unme bilkul same elements hon. Repeat aur order se farak nahi: FOLLOW aur WOLF ke letters se same set {F, O, L, W}.

1.6Subsetsउपसमुच्चयSubsets

A is a subset of B, A \subset B, if every element of A is also in B: a \in A \Rightarrow a \in B. Then B is a superset of A. If some element of A is not in B, write A \not\subset B. See the subset diagram.यदि A का हर अवयव B में भी हो तो A, B का उपसमुच्चय है: A \subset B, अर्थात् a \in A \Rightarrow a \in B। तब B, A का अधिसमुच्चय है। यदि A का कोई अवयव B में न हो तो A \not\subset B। उपसमुच्चय का चित्र देखिए।Agar A ka har element B mein bhi ho to A, B ka subset hai: A \subset B, yaani a \in A \Rightarrow a \in B. Tab B, A ka superset hai. Agar A ka koi element B mein na ho to A \not\subset B. Subset ka diagram dekhiye.

Subsets of \mathbf{R}: \mathbf{N} \subset \mathbf{Z} \subset \mathbf{Q} \subset \mathbf{R}; irrational numbers \mathbf{T} = \mathbf{R} - \mathbf{Q}, also a subset of \mathbf{R}.\mathbf{R} के उपसमुच्चय: \mathbf{N} \subset \mathbf{Z} \subset \mathbf{Q} \subset \mathbf{R}; अपरिमेय संख्याएँ \mathbf{T} = \mathbf{R} - \mathbf{Q}, यह भी \mathbf{R} का उपसमुच्चय है।\mathbf{R} ke subsets: \mathbf{N} \subset \mathbf{Z} \subset \mathbf{Q} \subset \mathbf{R}; irrational numbers \mathbf{T} = \mathbf{R} - \mathbf{Q}, ye bhi \mathbf{R} ka subset hai.

1.6.1Intervals as subsets of RR के उपसमुच्चय के रूप में अंतरालR ke subsets ke roop mein intervals

For real a < b, the intervals are: open (a, b) = \{x : a < x < b\}, closed [a, b] = \{x : a \le x \le b\}, half-open [a, b) = \{x : a \le x < b\} and (a, b] = \{x : a < x \le b\}. b - a is the length. Also [0, \infty) = non-negative reals, (-\infty, 0) = negative reals.वास्तविक a < b के लिए अंतराल: विवृत (a, b) = \{x : a < x < b\}, संवृत [a, b] = \{x : a \le x \le b\}, अर्ध-विवृत [a, b) = \{x : a \le x < b\} और (a, b] = \{x : a < x \le b\}। b - a अंतराल की लंबाई है। [0, \infty) = ऋणेतर वास्तविक संख्याएँ, (-\infty, 0) = ऋण वास्तविक संख्याएँ।Real a < b ke liye intervals: open (a, b) = \{x : a < x < b\}, closed [a, b] = \{x : a \le x \le b\}, half-open [a, b) = \{x : a \le x < b\} aur (a, b] = \{x : a < x \le b\}. b - a interval ki length hai. [0, \infty) = non-negative reals, (-\infty, 0) = negative reals.

Intervals on the real number lineवास्तविक संख्या रेखा पर अंतरालReal number line par intervalsSection 1.6.1 · 4 tappable partsअनुभाग 1.6.1 · 4 टैप करने लायक हिस्सेSection 1.6.1 · 4 tappable partsOpen

1.7Universal Setसार्वत्रिक समुच्चयUniversal Set

The universal set U is the basic set of a discussion: every set considered is its subset. For number systems \mathbf{R} can be U; for triangles, the set of all triangles in a plane. U is not unique — it depends on the context.सार्वत्रिक समुच्चय U किसी चर्चा का आधारभूत समुच्चय है: विचाराधीन हर समुच्चय उसका उपसमुच्चय है। संख्या पद्धतियों के लिए \mathbf{R}, त्रिभुजों के लिए तल के सभी त्रिभुजों का समुच्चय। U अद्वितीय नहीं — संदर्भ पर निर्भर है।Universal set U kisi discussion ka basic set hai: har set uska subset hota hai. Number systems ke liye \mathbf{R}, triangles ke liye plane ke saare triangles ka set. U unique nahi — context par depend karta hai.

1.8Venn Diagramsवेन आरेखVenn Diagrams

In a Venn diagram U is a rectangle and its subsets are circles inside it. Set operations become easy to *see* — tap the shaded regions in the Venn diagram picture.वेन आरेख में U को आयत और उसके उपसमुच्चयों को अंदर वृत्तों से दिखाते हैं। समुच्चय संक्रियाएँ आँखों से *दिखती* हैं — वेन आरेख चित्र के छायांकित भागों पर टैप कीजिए।Venn diagram mein U ko rectangle aur subsets ko andar circles se dikhate hain. Set operations aankhon se *dikhte* hain — Venn diagram picture ke shaded parts par tap kijiye.

Subset (A ⊂ B) and disjoint setsउपसमुच्चय (A ⊂ B) और असंयुक्त समुच्चयSubset (A ⊂ B) aur disjoint setsSections 1.6, 1.8 · 4 tappable partsअनुभाग 1.6, 1.8 · 4 टैप करने लायक हिस्सेSections 1.6, 1.8 · 4 tappable partsOpen

1.9Operations on Setsसमुच्चयों पर संक्रियाएँSets par operations

A \cup B = \{x : x \in A \text{ or } x \in B\}A \cup B = \{x : x \in A \text{ या } x \in B\}A \cup B = \{x : x \in A \text{ ya } x \in B\}
Union — all elements of A and of B (common ones once).सम्मिलन — A और B के सभी अवयव (उभयनिष्ठ एक बार)।Union — A aur B ke saare elements (common wale ek baar).
A \cap B = \{x : x \in A \text{ and } x \in B\}A \cap B = \{x : x \in A \text{ और } x \in B\}A \cap B = \{x : x \in A \text{ aur } x \in B\}
Intersection — only common elements. If A \cap B = \phi, A and B are disjoint.सर्वनिष्ठ — केवल उभयनिष्ठ अवयव। यदि A \cap B = \phi तो A, B असंयुक्त हैं।Intersection — sirf common elements. Agar A \cap B = \phi to A, B disjoint hain.
A - B = \{x : x \in A \text{ and } x \notin B\}A - B = \{x : x \in A \text{ और } x \notin B\}A - B = \{x : x \in A \text{ aur } x \notin B\}
Difference — elements of A not in B.अंतर — A के वे अवयव जो B में नहीं।Difference — A ke wo elements jo B mein nahi.

Example: A = \{1, 2, 3, 4, 5, 6\}, B = \{2, 3, 5, 7\}: A \cup B = \{1, \ldots, 7\}, A \cap B = \{2, 3, 5\}, A - B = \{1, 4, 6\}, B - A = \{7\}. The three sets A - B, A \cap B, B - A are mutually disjoint and together make A \cup B.उदाहरण: A = \{1, 2, 3, 4, 5, 6\}, B = \{2, 3, 5, 7\}: A \cup B = \{1, \ldots, 7\}, A \cap B = \{2, 3, 5\}, A - B = \{1, 4, 6\}, B - A = \{7\}। A - B, A \cap B, B - A परस्पर असंयुक्त हैं और मिलकर A \cup B बनाते हैं।Example: A = \{1, 2, 3, 4, 5, 6\}, B = \{2, 3, 5, 7\}: A \cup B = \{1, \ldots, 7\}, A \cap B = \{2, 3, 5\}, A - B = \{1, 4, 6\}, B - A = \{7\}. A - B, A \cap B, B - A aapas mein disjoint hain aur milkar A \cup B banate hain.

Venn diagrams: union, intersection, difference, complementवेन आरेख: सम्मिलन, सर्वनिष्ठ, अंतर, पूरकVenn diagrams: union, intersection, difference, complementSections 1.9, 1.10 · 4 tappable partsअनुभाग 1.9, 1.10 · 4 टैप करने लायक हिस्सेSections 1.9, 1.10 · 4 tappable partsOpen
Laws of union and intersectionCommutative: A \cup B = B \cup A, A \cap B = B \cap A. Associative: (A \cup B) \cup C = A \cup (B \cup C) (same for \cap). Identity: A \cup \phi = A, A \cap U = A. Idempotent: A \cup A = A \cap A = A. Distributive: A \cap (B \cup C) = (A \cap B) \cup (A \cap C), A \cup (B \cap C) = (A \cup B) \cap (A \cup C).सम्मिलन और सर्वनिष्ठ के नियमक्रमविनिमेय: A \cup B = B \cup A, A \cap B = B \cap A। साहचर्य: (A \cup B) \cup C = A \cup (B \cup C) (\cap के लिए भी)। तत्समक: A \cup \phi = A, A \cap U = A। वर्गसम: A \cup A = A \cap A = A। वितरण: A \cap (B \cup C) = (A \cap B) \cup (A \cap C), A \cup (B \cap C) = (A \cup B) \cap (A \cup C)।Union aur intersection ke lawsCommutative: A \cup B = B \cup A, A \cap B = B \cap A. Associative: (A \cup B) \cup C = A \cup (B \cup C) (\cap ke liye bhi). Identity: A \cup \phi = A, A \cap U = A. Idempotent: A \cup A = A \cap A = A. Distributive: A \cap (B \cup C) = (A \cap B) \cup (A \cap C), A \cup (B \cap C) = (A \cup B) \cap (A \cup C).
Example 15Let V = \{a, e, i, o, u\} and B = \{a, i, k, u\}. Find V - B and B - V.उदाहरण 15V = \{a, e, i, o, u\} और B = \{a, i, k, u\}। V - B और B - V ज्ञात कीजिए।Example 15V = \{a, e, i, o, u\} aur B = \{a, i, k, u\}. V - B aur B - V nikaliye.
Solution — step by step (1)हल — चरण दर चरण (1)Hal — step by step (1)
  1. Remove from V the letters also in B (a, i, u); then remove from B the letters also in V.V में से वे अक्षर हटाइए जो B में भी हैं (a, i, u); फिर B में से जो V में भी हैं।V mein se wo letters hataiye jo B mein bhi hain (a, i, u); phir B mein se jo V mein bhi hain.
    V - B = \{e, o\},\quad B - V = \{k\}

✔ V - B \ne B - V — order matters in a difference.V - B \ne B - V — अंतर में क्रम मायने रखता है।V - B \ne B - V — difference mein order matter karta hai.

1.10Complement of a Setसमुच्चय का पूरकSet ka Complement

For A \subset U, the complement of A is A' = \{x : x \in U,\ x \notin A\} = U - A. Example: U = \{1, \ldots, 10\}, A = \{1, 3, 5, 7, 9\} gives A' = \{2, 4, 6, 8, 10\}.A \subset U के लिए A का पूरक A' = \{x : x \in U,\ x \notin A\} = U - A है। उदाहरण: U = \{1, \ldots, 10\}, A = \{1, 3, 5, 7, 9\} से A' = \{2, 4, 6, 8, 10\}।A \subset U ke liye A ka complement A' = \{x : x \in U,\ x \notin A\} = U - A hai. Example: U = \{1, \ldots, 10\}, A = \{1, 3, 5, 7, 9\} se A' = \{2, 4, 6, 8, 10\}.

A \cup A' = U
Complement lawsपूरक नियमComplement laws
A \cap A' = \phi
(A \cup B)' = A' \cap B'
De Morgan’s lawsडी मॉर्गन के नियमDe Morgan ke laws
(A \cap B)' = A' \cup B'
(A')' = A
Double complement; complements of \phi and Uद्विपूरक; \phi और U के पूरकDouble complement; \phi aur U ke complement
\phi' = U
U' = \phi
Example 19U = \{1, 2, 3, 4, 5, 6\}, A = \{2, 3\}, B = \{3, 4, 5\}. Show that (A \cup B)' = A' \cap B'.उदाहरण 19U = \{1, 2, 3, 4, 5, 6\}, A = \{2, 3\}, B = \{3, 4, 5\}। दिखाइए कि (A \cup B)' = A' \cap B'।Example 19U = \{1, 2, 3, 4, 5, 6\}, A = \{2, 3\}, B = \{3, 4, 5\}. Dikhaiye ki (A \cup B)' = A' \cap B'.
Solution — step by step (2)हल — चरण दर चरण (2)Hal — step by step (2)
  1. A \cup B = \{2, 3, 4, 5\} \Rightarrow (A \cup B)' = \{1, 6\}
  2. A' = \{1, 4, 5, 6\},\ B' = \{1, 2, 6\} \Rightarrow A' \cap B' = \{1, 6\}

✔ Both sides are \{1, 6\} — verified.दोनों पक्ष \{1, 6\} — सत्यापित।Dono taraf \{1, 6\} — verify hua.

★ Yaad rakho — key formulas★ याद रखो — मुख्य सूत्र★ Yaad rakho — key formulas

  1. A set is a well-defined collection; order and repetition of elements don’t matter.समुच्चय सुपरिभाषित संग्रह है; अवयवों का क्रम और दोहराव मायने नहीं रखते।Set well-defined collection hai; elements ka order aur repeat matter nahi karta.
  2. \phi \subset A and A \subset A for every set A; n elements ⇒ 2^n subsets.हर समुच्चय A के लिए \phi \subset A और A \subset A; n अवयव ⇒ 2^n उपसमुच्चय।Har set A ke liye \phi \subset A aur A \subset A; n elements ⇒ 2^n subsets.
  3. A = B \iff A \subset B and B \subset A.A = B \iff A \subset B और B \subset A।A = B \iff A \subset B aur B \subset A.
  4. (a, b) excludes both ends, [a, b] includes both; length = b - a.(a, b) में दोनों सिरे शामिल नहीं, [a, b] में दोनों शामिल; लंबाई = b - a।(a, b) mein dono ends shaamil nahi, [a, b] mein dono; length = b - a.
  5. A - B = A \cap B'; in general A - B \ne B - A.A - B = A \cap B'; सामान्यतः A - B \ne B - A।A - B = A \cap B'; aam taur par A - B \ne B - A.
  6. De Morgan: (A \cup B)' = A' \cap B', (A \cap B)' = A' \cup B'.डी मॉर्गन: (A \cup B)' = A' \cap B', (A \cap B)' = A' \cup B'।De Morgan: (A \cup B)' = A' \cap B', (A \cap B)' = A' \cup B'.
  7. A \cup A' = U, A \cap A' = \phi, (A')' = A.A \cup A' = U, A \cap A' = \phi, (A')' = A।A \cup A' = U, A \cap A' = \phi, (A')' = A.

⚠ Common mistakes⚠ आम गलतियाँ⚠ Common mistakes

  • Writing \{3\} \in A when you mean \{3\} \subset A. An element belongs (\in); a set is contained (\subset).\{3\} \subset A की जगह \{3\} \in A लिखना। अवयव "का सदस्य" (\in) होता है; समुच्चय "का उपसमुच्चय" (\subset)।\{3\} \subset A ki jagah \{3\} \in A likhna. Element "belongs" (\in); set "contained" (\subset).
  • Thinking \phi = \{0\} or \phi = \{\phi\}. \{0\} and \{\phi\} each have one element; \phi has none.\phi = \{0\} या \phi = \{\phi\} मानना। \{0\} और \{\phi\} में एक-एक अवयव है; \phi में कोई नहीं।\phi = \{0\} ya \phi = \{\phi\} samajhna. \{0\} aur \{\phi\} mein ek-ek element hai; \phi mein koi nahi.
  • Repeating letters in roster form: letters of BETTER = {B, E, T, R}.रोस्टर रूप में अक्षर दोहराना: BETTER के अक्षर = {B, E, T, R}।Roster form mein letters repeat karna: BETTER ke letters = {B, E, T, R}.
  • Mixing up brackets: [-3, 0) includes −3 but not 0.कोष्ठक गड़बड़ करना: [-3, 0) में −3 शामिल है पर 0 नहीं।Brackets mix karna: [-3, 0) mein −3 shaamil hai par 0 nahi.
  • Taking 1 as prime or composite. 1 is neither — the complement of primes in N is "composite numbers and 1".1 को अभाज्य या भाज्य मानना। 1 दोनों नहीं — N में अभाज्य संख्याओं का पूरक "भाज्य संख्याएँ और 1" है।1 ko prime ya composite maanna. 1 dono nahi — N mein primes ka complement "composite numbers aur 1" hai.

✎ NCERT exercises — solutions✎ NCERT प्रश्नावली — हल✎ NCERT exercises — solutions

All 49 questions of the NCERT exercises. Tap a question to see the solution — try it yourself first!NCERT प्रश्नावली के सभी 49 प्रश्न। हल देखने के लिए प्रश्न पर टैप करें — पहले स्वयं हल करके देखें!NCERT exercises ke saare 49 questions. Hal dekhne ke liye question par tap kijiye — pehle khud try kijiye!

Exercise 1.1प्रश्नावली 1.1Exercise 1.1 · 6 Q

Q1. Which of these are sets? Justify. (i) Months of a year beginning with J (ii) Ten most talented writers of India (iii) Eleven best cricket batsmen of the world (iv) All boys in your class (v) Natural numbers less than 100 (vi) Novels written by Munshi Prem Chand (vii) All even integers (viii) Questions in this chapter (ix) Most dangerous animals of the worldइनमें कौन-से समुच्चय हैं? कारण दीजिए। (i) J से शुरू होने वाले वर्ष के महीने (ii) भारत के दस सबसे प्रतिभाशाली लेखक (iii) विश्व के ग्यारह सर्वश्रेष्ठ बल्लेबाज़ (iv) आपकी कक्षा के सभी बालक (v) 100 से कम प्राकृत संख्याएँ (vi) मुंशी प्रेमचंद के उपन्यास (vii) सभी सम पूर्णांक (viii) इस अध्याय के प्रश्न (ix) विश्व के सबसे खतरनाक जानवरWhich of these are sets? Justify. (i) Months of a year beginning with J (ii) Ten most talented writers of India (iii) Eleven best cricket batsmen of the world (iv) All boys in your class (v) Natural numbers less than 100 (vi) Novels written by Munshi Prem Chand (vii) All even integers (viii) Questions in this chapter (ix) Most dangerous animals of the world
  1. Opinion words ("most talented", "best", "most dangerous") make (ii), (iii), (ix) not well-defined.राय वाले शब्द ("सबसे प्रतिभाशाली", "सर्वश्रेष्ठ", "सबसे खतरनाक") के कारण (ii), (iii), (ix) सुपरिभाषित नहीं।Opinion wale words ("most talented", "best", "most dangerous") ki wajah se (ii), (iii), (ix) well-defined nahi.

✔ Sets: (i), (iv), (v), (vi), (vii), (viii). Not sets: (ii), (iii), (ix).समुच्चय: (i), (iv), (v), (vi), (vii), (viii)। समुच्चय नहीं: (ii), (iii), (ix)।Sets: (i), (iv), (v), (vi), (vii), (viii). Sets nahi: (ii), (iii), (ix).

Q2. A = \{1, 2, 3, 4, 5, 6\}. Put \in or \notin: (i) 5 … A (ii) 8 … A (iii) 0 … A (iv) 4 … A (v) 2 … A (vi) 10 … AA = \{1, 2, 3, 4, 5, 6\}। \in या \notin भरिए: (i) 5 … A (ii) 8 … A (iii) 0 … A (iv) 4 … A (v) 2 … A (vi) 10 … AA = \{1, 2, 3, 4, 5, 6\}. Put \in or \notin: (i) 5 … A (ii) 8 … A (iii) 0 … A (iv) 4 … A (v) 2 … A (vi) 10 … A

✔ (i) 5 \in A (ii) 8 \notin A (iii) 0 \notin A (iv) 4 \in A (v) 2 \in A (vi) 10 \notin A

Q3. Roster form: (i) \{x : x \in \mathbf{Z},\ -3 \le x < 7\} (ii) natural numbers less than 6 (iii) two-digit natural numbers whose digits add to 8 (iv) primes dividing 60 (v) letters of TRIGONOMETRY (vi) letters of BETTERरोस्टर रूप: (i) \{x : x \in \mathbf{Z},\ -3 \le x < 7\} (ii) 6 से कम प्राकृत संख्याएँ (iii) दो अंकों की प्राकृत संख्याएँ जिनके अंकों का योग 8 है (iv) 60 के अभाज्य भाजक (v) TRIGONOMETRY के अक्षर (vi) BETTER के अक्षरRoster form: (i) \{x : x \in \mathbf{Z},\ -3 \le x < 7\} (ii) natural numbers less than 6 (iii) two-digit natural numbers whose digits add to 8 (iv) primes dividing 60 (v) letters of TRIGONOMETRY (vi) letters of BETTER
  1. A = \{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6\},\quad B = \{1, 2, 3, 4, 5\}
  2. (iii) first digit 1 to 8, second digit = 8 − first. (iv) 60 = 2^2 \cdot 3 \cdot 5.(iii) पहला अंक 1 से 8, दूसरा = 8 − पहला। (iv) 60 = 2^2 \cdot 3 \cdot 5।(iii) pehla digit 1 se 8, doosra = 8 − pehla. (iv) 60 = 2^2 \cdot 3 \cdot 5.
    C = \{17, 26, 35, 44, 53, 62, 71, 80\},\quad D = \{2, 3, 5\}
  3. E = \{T, R, I, G, O, N, M, E, Y\},\quad F = \{B, E, T, R\}
Q4. Set-builder form: (i) {3, 6, 9, 12} (ii) {2, 4, 8, 16, 32} (iii) {5, 25, 125, 625} (iv) {2, 4, 6, …} (v) {1, 4, 9, …, 100}समुच्चय-निर्माण रूप: (i) {3, 6, 9, 12} (ii) {2, 4, 8, 16, 32} (iii) {5, 25, 125, 625} (iv) {2, 4, 6, …} (v) {1, 4, 9, …, 100}Set-builder form: (i) {3, 6, 9, 12} (ii) {2, 4, 8, 16, 32} (iii) {5, 25, 125, 625} (iv) {2, 4, 6, …} (v) {1, 4, 9, …, 100}
  1. (i)\ \{x : x = 3n,\ n \in \mathbf{N},\ 1 \le n \le 4\}\quad (ii)\ \{x : x = 2^n,\ n \in \mathbf{N},\ 1 \le n \le 5\}
  2. (iii)\ \{x : x = 5^n,\ n \in \mathbf{N},\ 1 \le n \le 4\}\quad (iv)\ \{x : x \text{ is an even natural number}\}(iii)\ \{x : x = 5^n,\ n \in \mathbf{N},\ 1 \le n \le 4\}\quad (iv)\ \{x : x \text{ एक सम प्राकृत संख्या है}\}(iii)\ \{x : x = 5^n,\ n \in \mathbf{N},\ 1 \le n \le 4\}\quad (iv)\ \{x : x \text{ ek even natural number hai}\}
  3. (v)\ \{x : x = n^2,\ n \in \mathbf{N},\ 1 \le n \le 10\}
Q5. List the elements: (i) odd natural numbers (ii) \{x : x \in \mathbf{Z},\ -\frac{1}{2} < x < \frac{9}{2}\} (iii) \{x : x \in \mathbf{Z},\ x^2 \le 4\} (iv) letters of LOYAL (v) months not having 31 days (vi) consonants before kअवयव लिखिए: (i) विषम प्राकृत संख्याएँ (ii) \{x : x \in \mathbf{Z},\ -\frac{1}{2} < x < \frac{9}{2}\} (iii) \{x : x \in \mathbf{Z},\ x^2 \le 4\} (iv) LOYAL के अक्षर (v) वे महीने जिनमें 31 दिन नहीं (vi) k से पहले के व्यंजनList the elements: (i) odd natural numbers (ii) \{x : x \in \mathbf{Z},\ -\frac{1}{2} < x < \frac{9}{2}\} (iii) \{x : x \in \mathbf{Z},\ x^2 \le 4\} (iv) letters of LOYAL (v) months not having 31 days (vi) consonants before k
  1. A = \{1, 3, 5, \ldots\},\quad B = \{0, 1, 2, 3, 4\},\quad C = \{-2, -1, 0, 1, 2\},\quad D = \{L, O, Y, A\}

✔ E = {February, April, June, September, November}; F = {b, c, d, f, g, h, j}E = {फ़रवरी, अप्रैल, जून, सितंबर, नवंबर}; F = {b, c, d, f, g, h, j}E = {February, April, June, September, November}; F = {b, c, d, f, g, h, j}

Q6. Match (i) {1, 2, 3, 6} (ii) {2, 3} (iii) {M, A, T, H, E, I, C, S} (iv) {1, 3, 5, 7, 9} with (a) primes dividing 6 (b) odd natural numbers below 10 (c) natural divisors of 6 (d) letters of MATHEMATICSमिलान कीजिए (i) {1, 2, 3, 6} (ii) {2, 3} (iii) {M, A, T, H, E, I, C, S} (iv) {1, 3, 5, 7, 9} को (a) 6 के अभाज्य भाजक (b) 10 से कम विषम प्राकृत संख्याएँ (c) 6 के प्राकृत भाजक (d) MATHEMATICS के अक्षरMatch (i) {1, 2, 3, 6} (ii) {2, 3} (iii) {M, A, T, H, E, I, C, S} (iv) {1, 3, 5, 7, 9} with (a) primes dividing 6 (b) odd natural numbers below 10 (c) natural divisors of 6 (d) letters of MATHEMATICS

✔ (i) ↔ (c), (ii) ↔ (a), (iii) ↔ (d), (iv) ↔ (b)

Exercise 1.2प्रश्नावली 1.2Exercise 1.2 · 6 Q

Q1. Which are null sets? (i) Odd natural numbers divisible by 2 (ii) Even primes (iii) \{x : x \in \mathbf{N},\ x < 5 \text{ and } x > 7\} (iv) Points common to two parallel linesकौन-से रिक्त समुच्चय हैं? (i) 2 से विभाज्य विषम प्राकृत संख्याएँ (ii) सम अभाज्य संख्याएँ (iii) \{x : x \in \mathbf{N},\ x < 5 \text{ और } x > 7\} (iv) दो समांतर रेखाओं के उभयनिष्ठ बिंदुWhich are null sets? (i) Odd natural numbers divisible by 2 (ii) Even primes (iii) \{x : x \in \mathbf{N},\ x < 5 \text{ and } x > 7\} (iv) Points common to two parallel lines
  1. (ii) = {2} is not empty. The others can have no element.(ii) = {2} रिक्त नहीं। बाकी में कोई अवयव संभव नहीं।(ii) = {2} empty nahi. Baaki mein koi element possible nahi.

✔ (i), (iii), (iv)

Q2. Finite or infinite? (i) Months of a year (ii) {1, 2, 3, …} (iii) {1, 2, …, 100} (iv) Positive integers greater than 100 (v) Primes less than 99परिमित या अपरिमित? (i) वर्ष के महीने (ii) {1, 2, 3, …} (iii) {1, 2, …, 100} (iv) 100 से बड़े धन पूर्णांक (v) 99 से कम अभाज्य संख्याएँFinite or infinite? (i) Months of a year (ii) {1, 2, 3, …} (iii) {1, 2, …, 100} (iv) Positive integers greater than 100 (v) Primes less than 99

✔ (i) finite (ii) infinite (iii) finite (iv) infinite (v) finite (25 primes)(i) परिमित (ii) अपरिमित (iii) परिमित (iv) अपरिमित (v) परिमित (25 अभाज्य)(i) finite (ii) infinite (iii) finite (iv) infinite (v) finite (25 primes)

Q3. Finite or infinite? (i) Lines parallel to the x-axis (ii) Letters of the English alphabet (iii) Multiples of 5 (iv) Animals living on the earth (v) Circles through the origin (0, 0)परिमित या अपरिमित? (i) x-अक्ष के समांतर रेखाएँ (ii) अंग्रेज़ी वर्णमाला के अक्षर (iii) 5 के गुणज (iv) पृथ्वी पर रहने वाले जानवर (v) मूलबिंदु (0, 0) से जाने वाले वृत्तFinite or infinite? (i) Lines parallel to the x-axis (ii) Letters of the English alphabet (iii) Multiples of 5 (iv) Animals living on the earth (v) Circles through the origin (0, 0)

✔ (i) infinite (ii) finite (iii) infinite (iv) finite (huge but countable) (v) infinite(i) अपरिमित (ii) परिमित (iii) अपरिमित (iv) परिमित (बहुत बड़ा पर गिनने योग्य) (v) अपरिमित(i) infinite (ii) finite (iii) infinite (iv) finite (bahut bada par ginne layak) (v) infinite

Q4. Is A = B? (i) {a, b, c, d}, {d, c, b, a} (ii) {4, 8, 12, 16}, {8, 4, 16, 18} (iii) {2, 4, 6, 8, 10}, {x : x positive even integer, x ≤ 10} (iv) {x : x multiple of 10}, {10, 15, 20, 25, 30, …}क्या A = B? (i) {a, b, c, d}, {d, c, b, a} (ii) {4, 8, 12, 16}, {8, 4, 16, 18} (iii) {2, 4, 6, 8, 10}, {x : x धन सम पूर्णांक, x ≤ 10} (iv) {x : x, 10 का गुणज}, {10, 15, 20, 25, 30, …}Is A = B? (i) {a, b, c, d}, {d, c, b, a} (ii) {4, 8, 12, 16}, {8, 4, 16, 18} (iii) {2, 4, 6, 8, 10}, {x : x positive even integer, x ≤ 10} (iv) {x : x multiple of 10}, {10, 15, 20, 25, 30, …}

✔ (i) Yes (ii) No: 12 ∈ A, 12 ∉ B (iii) Yes (iv) No: 15 ∈ B, 15 ∉ A(i) हाँ (ii) नहीं: 12 ∈ A, 12 ∉ B (iii) हाँ (iv) नहीं: 15 ∈ B, 15 ∉ A(i) Haan (ii) Nahi: 12 ∈ A, 12 ∉ B (iii) Haan (iv) Nahi: 15 ∈ B, 15 ∉ A

Q5. Are these equal? (i) A = {2, 3}, B = solutions of x^2 + 5x + 6 = 0 (ii) letters of FOLLOW, letters of WOLFक्या ये समान हैं? (i) A = {2, 3}, B = x^2 + 5x + 6 = 0 के हल (ii) FOLLOW के अक्षर, WOLF के अक्षरAre these equal? (i) A = {2, 3}, B = solutions of x^2 + 5x + 6 = 0 (ii) letters of FOLLOW, letters of WOLF
  1. x^2 + 5x + 6 = (x + 2)(x + 3) = 0 \Rightarrow B = \{-2, -3\} \ne A
  2. \{F, O, L, W\} = \{W, O, L, F\}

✔ (i) Not equal (ii) Equal(i) समान नहीं (ii) समान(i) Equal nahi (ii) Equal

Q6. Select equal sets: A = {2, 4, 8, 12}, B = {1, 2, 3, 4}, C = {4, 8, 12, 14}, D = {3, 1, 4, 2}, E = {−1, 1}, F = {0, a}, G = {1, −1}, H = {0, 1}समान समुच्चय चुनिए: A = {2, 4, 8, 12}, B = {1, 2, 3, 4}, C = {4, 8, 12, 14}, D = {3, 1, 4, 2}, E = {−1, 1}, F = {0, a}, G = {1, −1}, H = {0, 1}Select equal sets: A = {2, 4, 8, 12}, B = {1, 2, 3, 4}, C = {4, 8, 12, 14}, D = {3, 1, 4, 2}, E = {−1, 1}, F = {0, a}, G = {1, −1}, H = {0, 1}

✔ B = D, E = G

Exercise 1.3प्रश्नावली 1.3Exercise 1.3 · 8 Q

Q1. Fill \subset or \not\subset: (i) {2, 3, 4} … {1, 2, 3, 4, 5} (ii) {a, b, c} … {b, c, d} (iii) Class XI students of your school … students of your school (iv) circles in a plane … circles of radius 1 in that plane (v) triangles in a plane … rectangles in the plane (vi) equilateral triangles … triangles (vii) even natural numbers … integers\subset या \not\subset भरिए: (i) {2, 3, 4} … {1, 2, 3, 4, 5} (ii) {a, b, c} … {b, c, d} (iii) विद्यालय के कक्षा XI के विद्यार्थी … विद्यालय के विद्यार्थी (iv) तल के वृत्त … उसी तल में त्रिज्या 1 वाले वृत्त (v) तल के त्रिभुज … तल के आयत (vi) समबाहु त्रिभुज … त्रिभुज (vii) सम प्राकृत संख्याएँ … पूर्णांकFill \subset or \not\subset: (i) {2, 3, 4} … {1, 2, 3, 4, 5} (ii) {a, b, c} … {b, c, d} (iii) Class XI students of your school … students of your school (iv) circles in a plane … circles of radius 1 in that plane (v) triangles in a plane … rectangles in the plane (vi) equilateral triangles … triangles (vii) even natural numbers … integers

✔ (i) \subset (ii) \not\subset (iii) \subset (iv) \not\subset (v) \not\subset (vi) \subset (vii) \subset

Q2. True or false: (i) \{a, b\} \not\subset \{b, c, a\} (ii) {a, e} ⊂ {vowels of English} (iii) {1, 2, 3} ⊂ {1, 3, 5} (iv) {a} ⊂ {a, b, c} (v) {a} ∈ {a, b, c} (vi) {even natural numbers less than 6} ⊂ {natural numbers dividing 36}सत्य या असत्य: (i) \{a, b\} \not\subset \{b, c, a\} (ii) {a, e} ⊂ {अंग्रेज़ी स्वर} (iii) {1, 2, 3} ⊂ {1, 3, 5} (iv) {a} ⊂ {a, b, c} (v) {a} ∈ {a, b, c} (vi) {6 से कम सम प्राकृत संख्याएँ} ⊂ {36 के प्राकृत भाजक}True or false: (i) \{a, b\} \not\subset \{b, c, a\} (ii) {a, e} ⊂ {vowels of English} (iii) {1, 2, 3} ⊂ {1, 3, 5} (iv) {a} ⊂ {a, b, c} (v) {a} ∈ {a, b, c} (vi) {even natural numbers less than 6} ⊂ {natural numbers dividing 36}
  1. (v) {a} is a set, not an element of {a, b, c}. (vi) {2, 4}: both divide 36.(v) {a} एक समुच्चय है, {a, b, c} का अवयव नहीं। (vi) {2, 4}: दोनों 36 को विभाजित करते हैं।(v) {a} ek set hai, {a, b, c} ka element nahi. (vi) {2, 4}: dono 36 ko divide karte hain.

✔ (i) F (ii) T (iii) F (iv) T (v) F (vi) T

Q3. A = \{1, 2, \{3, 4\}, 5\}. Which are incorrect and why? (i) {3, 4} ⊂ A (ii) {3, 4} ∈ A (iii) {{3, 4}} ⊂ A (iv) 1 ∈ A (v) 1 ⊂ A (vi) {1, 2, 5} ⊂ A (vii) {1, 2, 5} ∈ A (viii) {1, 2, 3} ⊂ A (ix) φ ∈ A (x) φ ⊂ A (xi) {φ} ⊂ AA = \{1, 2, \{3, 4\}, 5\}। कौन-से कथन गलत हैं और क्यों? (i) {3, 4} ⊂ A (ii) {3, 4} ∈ A (iii) {{3, 4}} ⊂ A (iv) 1 ∈ A (v) 1 ⊂ A (vi) {1, 2, 5} ⊂ A (vii) {1, 2, 5} ∈ A (viii) {1, 2, 3} ⊂ A (ix) φ ∈ A (x) φ ⊂ A (xi) {φ} ⊂ AA = \{1, 2, \{3, 4\}, 5\}. Which are incorrect and why? (i) {3, 4} ⊂ A (ii) {3, 4} ∈ A (iii) {{3, 4}} ⊂ A (iv) 1 ∈ A (v) 1 ⊂ A (vi) {1, 2, 5} ⊂ A (vii) {1, 2, 5} ∈ A (viii) {1, 2, 3} ⊂ A (ix) φ ∈ A (x) φ ⊂ A (xi) {φ} ⊂ A
  1. A has exactly 4 elements: 1, 2, {3, 4}, 5. The numbers 3 and 4 are NOT elements of A.A के ठीक 4 अवयव हैं: 1, 2, {3, 4}, 5। संख्याएँ 3 और 4, A के अवयव नहीं हैं।A ke theek 4 elements hain: 1, 2, {3, 4}, 5. Numbers 3 aur 4, A ke elements nahi hain.
  2. (i) 3 ∉ A, so {3, 4} ⊄ A. (v) 1 is an element: write 1 ∈ A. (vii) {1, 2, 5} is a subset, not an element. (viii) 3 ∉ A. (ix) φ is not an element of A. (xi) {φ} ⊂ A needs φ ∈ A.(i) 3 ∉ A, इसलिए {3, 4} ⊄ A। (v) 1 अवयव है: 1 ∈ A लिखें। (vii) {1, 2, 5} उपसमुच्चय है, अवयव नहीं। (viii) 3 ∉ A। (ix) φ, A का अवयव नहीं। (xi) {φ} ⊂ A के लिए φ ∈ A चाहिए।(i) 3 ∉ A, isliye {3, 4} ⊄ A. (v) 1 element hai: 1 ∈ A likhiye. (vii) {1, 2, 5} subset hai, element nahi. (viii) 3 ∉ A. (ix) φ, A ka element nahi. (xi) {φ} ⊂ A ke liye φ ∈ A chahiye.

✔ Incorrect: (i), (v), (vii), (viii), (ix), (xi). Correct: (ii), (iii), (iv), (vi), (x).गलत: (i), (v), (vii), (viii), (ix), (xi)। सही: (ii), (iii), (iv), (vi), (x)।Galat: (i), (v), (vii), (viii), (ix), (xi). Sahi: (ii), (iii), (iv), (vi), (x).

Q4. Write all subsets of (i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) φसभी उपसमुच्चय लिखिए: (i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) φWrite all subsets of (i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) φ
  1. (i)\ \phi,\ \{a\}\qquad (ii)\ \phi,\ \{a\},\ \{b\},\ \{a, b\}
  2. (iii)\ \phi, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\qquad (iv)\ \phi

✔ Counts 2, 4, 8, 1 = 2^n.संख्या 2, 4, 8, 1 = 2^n।Ginti 2, 4, 8, 1 = 2^n.

Q5. Write as intervals: (i) \{x \in \mathbf{R} : -4 < x \le 6\} (ii) \{x \in \mathbf{R} : -12 < x < -10\} (iii) \{x \in \mathbf{R} : 0 \le x < 7\} (iv) \{x \in \mathbf{R} : 3 \le x \le 4\}अंतराल के रूप में लिखिए: (i) \{x \in \mathbf{R} : -4 < x \le 6\} (ii) \{x \in \mathbf{R} : -12 < x < -10\} (iii) \{x \in \mathbf{R} : 0 \le x < 7\} (iv) \{x \in \mathbf{R} : 3 \le x \le 4\}Write as intervals: (i) \{x \in \mathbf{R} : -4 < x \le 6\} (ii) \{x \in \mathbf{R} : -12 < x < -10\} (iii) \{x \in \mathbf{R} : 0 \le x < 7\} (iv) \{x \in \mathbf{R} : 3 \le x \le 4\}

✔ (i) (-4, 6] (ii) (-12, -10) (iii) [0, 7) (iv) [3, 4]

Q6. Set-builder form: (i) (−3, 0) (ii) [6, 12] (iii) (6, 12] (iv) [−23, 5)समुच्चय-निर्माण रूप: (i) (−3, 0) (ii) [6, 12] (iii) (6, 12] (iv) [−23, 5)Set-builder form: (i) (−3, 0) (ii) [6, 12] (iii) (6, 12] (iv) [−23, 5)
  1. \{x \in \mathbf{R} : -3 < x < 0\},\ \{x \in \mathbf{R} : 6 \le x \le 12\}
  2. \{x \in \mathbf{R} : 6 < x \le 12\},\ \{x \in \mathbf{R} : -23 \le x < 5\}
Q7. Propose universal set(s) for (i) the set of right triangles (ii) the set of isosceles triangles.इनके लिए सार्वत्रिक समुच्चय बताइए: (i) समकोण त्रिभुजों का समुच्चय (ii) समद्विबाहु त्रिभुजों का समुच्चय।Propose universal set(s) for (i) the set of right triangles (ii) the set of isosceles triangles.

✔ For both: the set of all triangles in the plane (or all polygons in the plane).दोनों के लिए: तल के सभी त्रिभुजों का समुच्चय (या तल के सभी बहुभुज)।Dono ke liye: plane ke saare triangles ka set (ya saare polygons).

Q8. A = {1, 3, 5}, B = {2, 4, 6}, C = {0, 2, 4, 6, 8}. Which can be a universal set for all three? (i) {0, 1, 2, 3, 4, 5, 6} (ii) φ (iii) {0, 1, 2, …, 10} (iv) {1, 2, …, 8}A = {1, 3, 5}, B = {2, 4, 6}, C = {0, 2, 4, 6, 8}। तीनों के लिए सार्वत्रिक समुच्चय कौन-सा हो सकता है? (i) {0, 1, 2, 3, 4, 5, 6} (ii) φ (iii) {0, 1, 2, …, 10} (iv) {1, 2, …, 8}A = {1, 3, 5}, B = {2, 4, 6}, C = {0, 2, 4, 6, 8}. Which can be a universal set for all three? (i) {0, 1, 2, 3, 4, 5, 6} (ii) φ (iii) {0, 1, 2, …, 10} (iv) {1, 2, …, 8}
  1. U must contain A \cup B \cup C = \{0, 1, 2, 3, 4, 5, 6, 8\}; (i) lacks 8, (iv) lacks 0.U में A \cup B \cup C = \{0, 1, 2, 3, 4, 5, 6, 8\} होना चाहिए; (i) में 8 नहीं, (iv) में 0 नहीं।U mein A \cup B \cup C = \{0, 1, 2, 3, 4, 5, 6, 8\} hona chahiye; (i) mein 8 nahi, (iv) mein 0 nahi.

✔ Only (iii)केवल (iii)Sirf (iii)

Exercise 1.4प्रश्नावली 1.4Exercise 1.4 · 12 Q

Q1. Find the union: (i) X = {1, 3, 5}, Y = {1, 2, 3} (ii) A = {a, e, i, o, u}, B = {a, b, c} (iii) A = natural multiples of 3, B = natural numbers less than 6 (iv) A = {x ∈ N : 1 < x ≤ 6}, B = {x ∈ N : 6 < x < 10} (v) A = {1, 2, 3}, B = φसम्मिलन ज्ञात कीजिए: (i) X = {1, 3, 5}, Y = {1, 2, 3} (ii) A = {a, e, i, o, u}, B = {a, b, c} (iii) A = 3 के प्राकृत गुणज, B = 6 से कम प्राकृत संख्याएँ (iv) A = {x ∈ N : 1 < x ≤ 6}, B = {x ∈ N : 6 < x < 10} (v) A = {1, 2, 3}, B = φFind the union: (i) X = {1, 3, 5}, Y = {1, 2, 3} (ii) A = {a, e, i, o, u}, B = {a, b, c} (iii) A = natural multiples of 3, B = natural numbers less than 6 (iv) A = {x ∈ N : 1 < x ≤ 6}, B = {x ∈ N : 6 < x < 10} (v) A = {1, 2, 3}, B = φ
  1. (i)\ \{1, 2, 3, 5\}\quad (ii)\ \{a, b, c, e, i, o, u\}
  2. (iii)\ \{1, 2, 3, 4, 5, 6, 9, 12, 15, \ldots\}
  3. (iv)\ \{2, 3, \ldots, 9\} = \{x \in \mathbf{N} : 1 < x < 10\}\quad (v)\ \{1, 2, 3\}
Q2. A = {a, b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?A = {a, b}, B = {a, b, c}। क्या A ⊂ B? A ∪ B क्या है?A = {a, b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?

✔ Yes; A \cup B = \{a, b, c\} = B.हाँ; A \cup B = \{a, b, c\} = B।Haan; A \cup B = \{a, b, c\} = B.

Q3. If A \subset B, what is A \cup B?यदि A \subset B, तो A \cup B क्या है?If A \subset B, what is A \cup B?
  1. Every element of A is already in B.A का हर अवयव पहले से B में है।A ka har element pehle se B mein hai.

✔ A \cup B = B

Q4. A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8}, D = {7, 8, 9, 10}. Find (i) A∪B (ii) A∪C (iii) B∪C (iv) B∪D (v) A∪B∪C (vi) A∪B∪D (vii) B∪C∪DA = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8}, D = {7, 8, 9, 10}। ज्ञात कीजिए (i) A∪B (ii) A∪C (iii) B∪C (iv) B∪D (v) A∪B∪C (vi) A∪B∪D (vii) B∪C∪DA = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8}, D = {7, 8, 9, 10}. Find (i) A∪B (ii) A∪C (iii) B∪C (iv) B∪D (v) A∪B∪C (vi) A∪B∪D (vii) B∪C∪D
  1. (i)\ \{1,\ldots,6\}\quad (ii)\ \{1,\ldots,8\}\quad (iii)\ \{3,\ldots,8\}\quad (iv)\ \{3,\ldots,10\}
  2. (v)\ \{1,\ldots,8\}\quad (vi)\ \{1,\ldots,10\}\quad (vii)\ \{3,\ldots,10\}
Q5. Find the intersection of each pair of sets in Q1.प्रश्न 1 के प्रत्येक युग्म का सर्वनिष्ठ ज्ञात कीजिए।Find the intersection of each pair of sets in Q1.
  1. (iii) the only multiple of 3 below 6 is 3. (iv) no natural number is both ≤ 6 and > 6.(iii) 6 से कम 3 का एकमात्र गुणज 3 है। (iv) कोई संख्या ≤ 6 और > 6 दोनों नहीं।(iii) 6 se kam 3 ka ek hi multiple 3 hai. (iv) koi number ≤ 6 aur > 6 dono nahi.

✔ (i) {1, 3} (ii) {a} (iii) {3} (iv) φ (v) φ

Q6. A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}, D = {15, 17}. Find (i) A∩B (ii) B∩C (iii) A∩C∩D (iv) A∩C (v) B∩D (vi) A∩(B∪C) (vii) A∩D (viii) A∩(B∪D) (ix) (A∩B)∩(B∪C) (x) (A∪D)∩(B∪C)A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}, D = {15, 17}। ज्ञात कीजिए (i) A∩B (ii) B∩C (iii) A∩C∩D (iv) A∩C (v) B∩D (vi) A∩(B∪C) (vii) A∩D (viii) A∩(B∪D) (ix) (A∩B)∩(B∪C) (x) (A∪D)∩(B∪C)A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}, D = {15, 17}. Find (i) A∩B (ii) B∩C (iii) A∩C∩D (iv) A∩C (v) B∩D (vi) A∩(B∪C) (vii) A∩D (viii) A∩(B∪D) (ix) (A∩B)∩(B∪C) (x) (A∪D)∩(B∪C)
  1. B \cup C = \{7, 9, 11, 13, 15\},\ B \cup D = \{7, 9, 11, 13, 15, 17\},\ A \cup D = \{3, 5, 7, 9, 11, 15, 17\}
  2. (i)\ \{7,9,11\}\ (ii)\ \{11,13\}\ (iii)\ \phi\ (iv)\ \{11\}\ (v)\ \phi
  3. (vi)\ \{7,9,11\}\ (vii)\ \phi\ (viii)\ \{7,9,11\}\ (ix)\ \{7,9,11\}\ (x)\ \{7,9,11,15\}
Q7. A = N, B = even natural numbers, C = odd natural numbers, D = primes. Find (i) A∩B (ii) A∩C (iii) A∩D (iv) B∩C (v) B∩D (vi) C∩DA = N, B = सम प्राकृत संख्याएँ, C = विषम प्राकृत संख्याएँ, D = अभाज्य संख्याएँ। ज्ञात कीजिए (i) A∩B (ii) A∩C (iii) A∩D (iv) B∩C (v) B∩D (vi) C∩DA = N, B = even natural numbers, C = odd natural numbers, D = primes. Find (i) A∩B (ii) A∩C (iii) A∩D (iv) B∩C (v) B∩D (vi) C∩D

✔ (i) B (ii) C (iii) D (iv) φ (v) {2} (vi) {x : x is an odd prime}(i) B (ii) C (iii) D (iv) φ (v) {2} (vi) {x : x विषम अभाज्य संख्या है}(i) B (ii) C (iii) D (iv) φ (v) {2} (vi) {x : x odd prime hai}

Q8. Which pairs are disjoint? (i) {1, 2, 3, 4} and {x ∈ N : 4 ≤ x ≤ 6} (ii) {a, e, i, o, u} and {c, d, e, f} (iii) even integers and odd integersकौन-से युग्म असंयुक्त हैं? (i) {1, 2, 3, 4} और {x ∈ N : 4 ≤ x ≤ 6} (ii) {a, e, i, o, u} और {c, d, e, f} (iii) सम पूर्णांक और विषम पूर्णांकWhich pairs are disjoint? (i) {1, 2, 3, 4} and {x ∈ N : 4 ≤ x ≤ 6} (ii) {a, e, i, o, u} and {c, d, e, f} (iii) even integers and odd integers
  1. (i) 4 is common; (ii) e is common.(i) 4 उभयनिष्ठ; (ii) e उभयनिष्ठ।(i) 4 common; (ii) e common.

✔ Only (iii)केवल (iii)Sirf (iii)

Q9. A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}. Find (i) A−B (ii) A−C (iii) A−D (iv) B−A (v) C−A (vi) D−A (vii) B−C (viii) B−D (ix) C−B (x) D−B (xi) C−D (xii) D−CA = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}। ज्ञात कीजिए (i) A−B (ii) A−C (iii) A−D (iv) B−A (v) C−A (vi) D−A (vii) B−C (viii) B−D (ix) C−B (x) D−B (xi) C−D (xii) D−CA = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}. Find (i) A−B (ii) A−C (iii) A−D (iv) B−A (v) C−A (vi) D−A (vii) B−C (viii) B−D (ix) C−B (x) D−B (xi) C−D (xii) D−C
  1. (i)\ \{3,6,9,15,18,21\}\ (ii)\ \{3,9,15,18,21\}\ (iii)\ \{3,6,9,12,18,21\}
  2. (iv)\ \{4,8,16,20\}\ (v)\ \{2,4,8,10,14,16\}\ (vi)\ \{5,10,20\}
  3. (vii)\ \{20\}\ (viii)\ \{4,8,12,16\}\ (ix)\ \{2,6,10,14\}
  4. (x)\ \{5,10,15\}\ (xi)\ \{2,4,6,8,12,14,16\}\ (xii)\ \{5,15,20\}
Q10. X = {a, b, c, d}, Y = {f, b, d, g}. Find (i) X − Y (ii) Y − X (iii) X ∩ YX = {a, b, c, d}, Y = {f, b, d, g}। ज्ञात कीजिए (i) X − Y (ii) Y − X (iii) X ∩ YX = {a, b, c, d}, Y = {f, b, d, g}. Find (i) X − Y (ii) Y − X (iii) X ∩ Y

✔ (i) {a, c} (ii) {f, g} (iii) {b, d}

Q11. R = real numbers, Q = rational numbers. What is R − Q?R = वास्तविक संख्याएँ, Q = परिमेय संख्याएँ। R − Q क्या है?R = real numbers, Q = rational numbers. What is R − Q?

✔ The set of irrational numbers.अपरिमेय संख्याओं का समुच्चय।Irrational numbers ka set.

Q12. True or false? Justify. (i) {2, 3, 4, 5} and {3, 6} are disjoint (ii) {a, e, i, o, u} and {a, b, c, d} are disjoint (iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint (iv) {2, 6, 10} and {3, 7, 11} are disjointसत्य या असत्य? कारण दीजिए। (i) {2, 3, 4, 5} और {3, 6} असंयुक्त हैं (ii) {a, e, i, o, u} और {a, b, c, d} असंयुक्त हैं (iii) {2, 6, 10, 14} और {3, 7, 11, 15} असंयुक्त हैं (iv) {2, 6, 10} और {3, 7, 11} असंयुक्त हैंTrue or false? Justify. (i) {2, 3, 4, 5} and {3, 6} are disjoint (ii) {a, e, i, o, u} and {a, b, c, d} are disjoint (iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint (iv) {2, 6, 10} and {3, 7, 11} are disjoint

✔ (i) False (3 is common) (ii) False (a is common) (iii) True (iv) True(i) असत्य (3 उभयनिष्ठ) (ii) असत्य (a उभयनिष्ठ) (iii) सत्य (iv) सत्य(i) False (3 common) (ii) False (a common) (iii) True (iv) True

Exercise 1.5प्रश्नावली 1.5Exercise 1.5 · 7 Q

Q1. U = {1, 2, …, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8}, C = {3, 4, 5, 6}. Find (i) A′ (ii) B′ (iii) (A ∪ C)′ (iv) (A ∪ B)′ (v) (A′)′ (vi) (B − C)′U = {1, 2, …, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8}, C = {3, 4, 5, 6}। ज्ञात कीजिए (i) A′ (ii) B′ (iii) (A ∪ C)′ (iv) (A ∪ B)′ (v) (A′)′ (vi) (B − C)′U = {1, 2, …, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8}, C = {3, 4, 5, 6}. Find (i) A′ (ii) B′ (iii) (A ∪ C)′ (iv) (A ∪ B)′ (v) (A′)′ (vi) (B − C)′
  1. (i)\ \{5,6,7,8,9\}\quad (ii)\ \{1,3,5,7,9\}
  2. A \cup C = \{1,\ldots,6\} \Rightarrow (iii)\ \{7,8,9\};\quad A \cup B = \{1,2,3,4,6,8\} \Rightarrow (iv)\ \{5,7,9\}
  3. (v)\ A = \{1,2,3,4\};\quad B - C = \{2, 8\} \Rightarrow (vi)\ \{1,3,4,5,6,7,9\}
Q2. U = {a, b, c, d, e, f, g, h}. Complements of (i) A = {a, b, c} (ii) B = {d, e, f, g} (iii) C = {a, c, e, g} (iv) D = {f, g, h, a}U = {a, b, c, d, e, f, g, h}। पूरक लिखिए: (i) A = {a, b, c} (ii) B = {d, e, f, g} (iii) C = {a, c, e, g} (iv) D = {f, g, h, a}U = {a, b, c, d, e, f, g, h}. Complements of (i) A = {a, b, c} (ii) B = {d, e, f, g} (iii) C = {a, c, e, g} (iv) D = {f, g, h, a}

✔ (i) {d, e, f, g, h} (ii) {a, b, c, h} (iii) {b, d, f, h} (iv) {b, c, d, e}

Q3. With N as the universal set, complements of: (i) even natural numbers (ii) odd natural numbers (iii) positive multiples of 3 (iv) primes (v) natural numbers divisible by 3 and 5 (vi) perfect squares (vii) perfect cubes (viii) {x : x + 5 = 8} (ix) {x : 2x + 5 = 9} (x) {x : x ≥ 7} (xi) {x ∈ N : 2x + 1 > 10}N को सार्वत्रिक समुच्चय मानकर पूरक: (i) सम प्राकृत संख्याएँ (ii) विषम प्राकृत संख्याएँ (iii) 3 के धन गुणज (iv) अभाज्य संख्याएँ (v) 3 और 5 से विभाज्य प्राकृत संख्याएँ (vi) पूर्ण वर्ग (vii) पूर्ण घन (viii) {x : x + 5 = 8} (ix) {x : 2x + 5 = 9} (x) {x : x ≥ 7} (xi) {x ∈ N : 2x + 1 > 10}With N as the universal set, complements of: (i) even natural numbers (ii) odd natural numbers (iii) positive multiples of 3 (iv) primes (v) natural numbers divisible by 3 and 5 (vi) perfect squares (vii) perfect cubes (viii) {x : x + 5 = 8} (ix) {x : 2x + 5 = 9} (x) {x : x ≥ 7} (xi) {x ∈ N : 2x + 1 > 10}
  1. (i) odd natural numbers (ii) even natural numbers (iii) natural numbers that are not multiples of 3 (iv) composite numbers and 1 (v) natural numbers not divisible by 15, i.e. not divisible by 3 or not by 5 (vi) natural numbers that are not perfect squares (vii) natural numbers that are not perfect cubes(i) विषम प्राकृत संख्याएँ (ii) सम प्राकृत संख्याएँ (iii) वे प्राकृत संख्याएँ जो 3 की गुणज नहीं (iv) भाज्य संख्याएँ और 1 (v) वे प्राकृत संख्याएँ जो 15 से विभाज्य नहीं, अर्थात् 3 से या 5 से विभाज्य नहीं (vi) वे प्राकृत संख्याएँ जो पूर्ण वर्ग नहीं (vii) वे जो पूर्ण घन नहीं(i) odd natural numbers (ii) even natural numbers (iii) wo natural numbers jo 3 ke multiple nahi (iv) composite numbers aur 1 (v) wo natural numbers jo 15 se divisible nahi, yaani 3 se ya 5 se divisible nahi (vi) jo perfect square nahi (vii) jo perfect cube nahi
  2. (viii) x = 3 (ix) x = 2 (x) x ≥ 7 (xi) 2x + 1 > 10 ⇔ x > 4.5
    (viii)\ \{x \in \mathbf{N} : x \ne 3\}\ (ix)\ \{x \in \mathbf{N} : x \ne 2\}\ (x)\ \{x \in \mathbf{N} : x < 7\}\ (xi)\ \{x \in \mathbf{N} : x \le \tfrac{9}{2}\} = \{1,2,3,4\}
Q4. U = {1, …, 9}, A = {2, 4, 6, 8}, B = {2, 3, 5, 7}. Verify (i) (A ∪ B)′ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′U = {1, …, 9}, A = {2, 4, 6, 8}, B = {2, 3, 5, 7}। सत्यापित कीजिए (i) (A ∪ B)′ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′U = {1, …, 9}, A = {2, 4, 6, 8}, B = {2, 3, 5, 7}. Verify (i) (A ∪ B)′ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′
  1. A' = \{1,3,5,7,9\},\quad B' = \{1,4,6,8,9\}
  2. (i)\ A \cup B = \{2,3,4,5,6,7,8\} \Rightarrow (A \cup B)' = \{1, 9\} = A' \cap B'
  3. (ii)\ A \cap B = \{2\} \Rightarrow (A \cap B)' = \{1,3,4,5,6,7,8,9\} = A' \cup B'

✔ Both verified.दोनों सत्यापित।Dono verify hue.

Q5. Draw Venn diagrams for (i) (A ∪ B)′ (ii) A′ ∩ B′ (iii) (A ∩ B)′ (iv) A′ ∪ B′वेन आरेख बनाइए: (i) (A ∪ B)′ (ii) A′ ∩ B′ (iii) (A ∩ B)′ (iv) A′ ∪ B′Draw Venn diagrams for (i) (A ∪ B)′ (ii) A′ ∩ B′ (iii) (A ∩ B)′ (iv) A′ ∪ B′
  1. (i) and (ii): shade the part of the rectangle OUTSIDE both circles (they are the same region — De Morgan). (iii) and (iv): shade everything EXCEPT the overlap of the two circles. See the Venn diagram picture.(i) और (ii): आयत का वह भाग छायांकित करें जो दोनों वृत्तों के बाहर है (दोनों एक ही क्षेत्र — डी मॉर्गन)। (iii) और (iv): दोनों वृत्तों के उभयनिष्ठ भाग को छोड़कर सब छायांकित करें। वेन आरेख चित्र देखिए।(i) aur (ii): rectangle ka wo hissa shade kijiye jo dono circles ke bahar hai (dono same region — De Morgan). (iii) aur (iv): dono circles ke overlap ko chhod kar sab shade kijiye. Venn diagram picture dekhiye.
Q6. U = all triangles in a plane, A = triangles with at least one angle different from 60°. What is A′?U = तल के सभी त्रिभुज, A = वे त्रिभुज जिनका कम से कम एक कोण 60° से भिन्न है। A′ क्या है?U = all triangles in a plane, A = triangles with at least one angle different from 60°. What is A′?
  1. Not in A means every angle is exactly 60°.A में न होने का अर्थ है हर कोण ठीक 60° है।A mein na hone ka matlab hai har angle theek 60° hai.

✔ A′ = the set of all equilateral triangles.A′ = सभी समबाहु त्रिभुजों का समुच्चय।A′ = saare equilateral triangles ka set.

Q7. Fill in: (i) A ∪ A′ = … (ii) φ′ ∩ A = … (iii) A ∩ A′ = … (iv) U′ ∩ A = …रिक्त स्थान भरिए: (i) A ∪ A′ = … (ii) φ′ ∩ A = … (iii) A ∩ A′ = … (iv) U′ ∩ A = …Fill in: (i) A ∪ A′ = … (ii) φ′ ∩ A = … (iii) A ∩ A′ = … (iv) U′ ∩ A = …
  1. φ′ = U and U′ = φ.φ′ = U और U′ = φ।φ′ = U aur U′ = φ.

✔ (i) U (ii) U ∩ A = A (iii) φ (iv) φ ∩ A = φ

Miscellaneous Exercise on Chapter 1अध्याय 1 पर विविध प्रश्नावलीMiscellaneous Exercise on Chapter 1 · 10 Q

Q1. Which of these are subsets of one another? A = {x ∈ R : x² − 8x + 12 = 0}, B = {2, 4, 6}, C = {2, 4, 6, 8, …}, D = {6}इनमें कौन किसका उपसमुच्चय है? A = {x ∈ R : x² − 8x + 12 = 0}, B = {2, 4, 6}, C = {2, 4, 6, 8, …}, D = {6}Which of these are subsets of one another? A = {x ∈ R : x² − 8x + 12 = 0}, B = {2, 4, 6}, C = {2, 4, 6, 8, …}, D = {6}
  1. x^2 - 8x + 12 = (x - 2)(x - 6) = 0 \Rightarrow A = \{2, 6\}

✔ A \subset B, A \subset C, B \subset C, D \subset A, D \subset B, D \subset C

Q2. True or false? If true prove it, if false give an example. (i) x ∈ A and A ∈ B ⇒ x ∈ B (ii) A ⊂ B and B ∈ C ⇒ A ∈ C (iii) A ⊂ B and B ⊂ C ⇒ A ⊂ C (iv) A ⊄ B and B ⊄ C ⇒ A ⊄ C (v) x ∈ A and A ⊄ B ⇒ x ∈ B (vi) A ⊂ B and x ∉ B ⇒ x ∉ Aसत्य या असत्य? सत्य हो तो सिद्ध कीजिए, असत्य हो तो उदाहरण दीजिए। (i) x ∈ A और A ∈ B ⇒ x ∈ B (ii) A ⊂ B और B ∈ C ⇒ A ∈ C (iii) A ⊂ B और B ⊂ C ⇒ A ⊂ C (iv) A ⊄ B और B ⊄ C ⇒ A ⊄ C (v) x ∈ A और A ⊄ B ⇒ x ∈ B (vi) A ⊂ B और x ∉ B ⇒ x ∉ ATrue or false? If true prove it, if false give an example. (i) x ∈ A and A ∈ B ⇒ x ∈ B (ii) A ⊂ B and B ∈ C ⇒ A ∈ C (iii) A ⊂ B and B ⊂ C ⇒ A ⊂ C (iv) A ⊄ B and B ⊄ C ⇒ A ⊄ C (v) x ∈ A and A ⊄ B ⇒ x ∈ B (vi) A ⊂ B and x ∉ B ⇒ x ∉ A
  1. (i) False: A = {1}, B = {{1}, 2}: 1 ∈ A, A ∈ B, but 1 ∉ B.(i) असत्य: A = {1}, B = {{1}, 2}: 1 ∈ A, A ∈ B, पर 1 ∉ B।(i) False: A = {1}, B = {{1}, 2}: 1 ∈ A, A ∈ B, par 1 ∉ B.
  2. (ii) False: A = {1}, B = {1, 2}, C = {{1, 2}, 3}: A ⊂ B, B ∈ C, but A ∉ C.(ii) असत्य: A = {1}, B = {1, 2}, C = {{1, 2}, 3}: A ⊂ B, B ∈ C, पर A ∉ C।(ii) False: A = {1}, B = {1, 2}, C = {{1, 2}, 3}: A ⊂ B, B ∈ C, par A ∉ C.
  3. (iii) True: x ∈ A ⇒ x ∈ B (A ⊂ B) ⇒ x ∈ C (B ⊂ C).(iii) सत्य: x ∈ A ⇒ x ∈ B (A ⊂ B) ⇒ x ∈ C (B ⊂ C)।(iii) True: x ∈ A ⇒ x ∈ B (A ⊂ B) ⇒ x ∈ C (B ⊂ C).
  4. (iv) False: A = {1, 2}, B = {2, 3}, C = {1, 2, 4}: A ⊄ B, B ⊄ C, but A ⊂ C.(iv) असत्य: A = {1, 2}, B = {2, 3}, C = {1, 2, 4}: A ⊄ B, B ⊄ C, पर A ⊂ C।(iv) False: A = {1, 2}, B = {2, 3}, C = {1, 2, 4}: A ⊄ B, B ⊄ C, par A ⊂ C.
  5. (v) False: A = {1, 2}, B = {2}, x = 1: x ∈ A, A ⊄ B, but x ∉ B.(v) असत्य: A = {1, 2}, B = {2}, x = 1: x ∈ A, A ⊄ B, पर x ∉ B।(v) False: A = {1, 2}, B = {2}, x = 1: x ∈ A, A ⊄ B, par x ∉ B.
  6. (vi) True: if x ∈ A then x ∈ B (A ⊂ B), contradicting x ∉ B.(vi) सत्य: यदि x ∈ A तो x ∈ B (A ⊂ B), जो x ∉ B का खंडन है।(vi) True: agar x ∈ A to x ∈ B (A ⊂ B), jo x ∉ B ke khilaaf hai.

✔ (i) F (ii) F (iii) T (iv) F (v) F (vi) T

Q3. If A ∪ B = A ∪ C and A ∩ B = A ∩ C, show that B = C.यदि A ∪ B = A ∪ C और A ∩ B = A ∩ C, तो दिखाइए कि B = C।If A ∪ B = A ∪ C and A ∩ B = A ∩ C, show that B = C.
  1. B = B \cap (A \cup B) = B \cap (A \cup C) = (B \cap A) \cup (B \cap C)
  2. = (A \cap C) \cup (B \cap C) = (A \cup B) \cap C = (A \cup C) \cap C = C

✔ Hence B = C (distributive law used twice).अतः B = C (वितरण नियम दो बार लगाया)।Isliye B = C (distributive law do baar lagaya).

Q4. Show that these are equivalent: (i) A ⊂ B (ii) A − B = φ (iii) A ∪ B = B (iv) A ∩ B = Aदिखाइए कि ये तुल्य हैं: (i) A ⊂ B (ii) A − B = φ (iii) A ∪ B = B (iv) A ∩ B = AShow that these are equivalent: (i) A ⊂ B (ii) A − B = φ (iii) A ∪ B = B (iv) A ∩ B = A
  1. (i)⇒(ii): every x ∈ A is in B, so no element of A is outside B: A − B = φ.(i)⇒(ii): A का हर x, B में है, अतः A का कोई अवयव B के बाहर नहीं: A − B = φ।(i)⇒(ii): A ka har x, B mein hai, isliye A ka koi element B ke bahar nahi: A − B = φ.
  2. (ii)⇒(iii): A − B = φ means every element of A is in B, so A ∪ B adds nothing to B: A ∪ B = B.(ii)⇒(iii): A − B = φ का अर्थ है A का हर अवयव B में है, अतः A ∪ B = B।(ii)⇒(iii): A − B = φ ka matlab A ka har element B mein hai, isliye A ∪ B = B.
  3. (iii)⇒(iv): A ⊂ A ∪ B = B, so A ∩ B = A. (iv)⇒(i): A = A ∩ B ⊂ B.(iii)⇒(iv): A ⊂ A ∪ B = B, अतः A ∩ B = A। (iv)⇒(i): A = A ∩ B ⊂ B।(iii)⇒(iv): A ⊂ A ∪ B = B, isliye A ∩ B = A. (iv)⇒(i): A = A ∩ B ⊂ B.

✔ The chain (i)⇒(ii)⇒(iii)⇒(iv)⇒(i) closes, so all four are equivalent.श्रृंखला (i)⇒(ii)⇒(iii)⇒(iv)⇒(i) पूरी होती है, अतः चारों तुल्य हैं।Chain (i)⇒(ii)⇒(iii)⇒(iv)⇒(i) poori hoti hai, isliye chaaron equivalent hain.

Q5. Show that if A ⊂ B, then C − B ⊂ C − A.दिखाइए कि यदि A ⊂ B, तो C − B ⊂ C − A।Show that if A ⊂ B, then C − B ⊂ C − A.
  1. Take x ∈ C − B: x ∈ C and x ∉ B. Since A ⊂ B, x ∉ A too. So x ∈ C − A.x ∈ C − B लीजिए: x ∈ C और x ∉ B। चूँकि A ⊂ B, x ∉ A भी। अतः x ∈ C − A।x ∈ C − B lijiye: x ∈ C aur x ∉ B. Kyunki A ⊂ B, x ∉ A bhi. Isliye x ∈ C − A.

✔ C - B \subset C - A

Q6. Show that for any sets A and B, A = (A ∩ B) ∪ (A − B) and A ∪ (B − A) = A ∪ B.दिखाइए कि किन्हीं समुच्चयों A, B के लिए A = (A ∩ B) ∪ (A − B) और A ∪ (B − A) = A ∪ B।Show that for any sets A and B, A = (A ∩ B) ∪ (A − B) and A ∪ (B − A) = A ∪ B.
  1. (A \cap B) \cup (A \cap B') = A \cap (B \cup B') = A \cap U = A
  2. A \cup (B \cap A') = (A \cup B) \cap (A \cup A') = (A \cup B) \cap U = A \cup B

✔ Both proved (using X - Y = X \cap Y' and the distributive law).दोनों सिद्ध (X - Y = X \cap Y' और वितरण नियम से)।Dono prove hue (X - Y = X \cap Y' aur distributive law se).

Q7. Using properties of sets, show (i) A ∪ (A ∩ B) = A (ii) A ∩ (A ∪ B) = Aसमुच्चयों के गुणों से दिखाइए (i) A ∪ (A ∩ B) = A (ii) A ∩ (A ∪ B) = AUsing properties of sets, show (i) A ∪ (A ∩ B) = A (ii) A ∩ (A ∪ B) = A
  1. (i)\ A \cup (A \cap B) = (A \cap U) \cup (A \cap B) = A \cap (U \cup B) = A \cap U = A
  2. (ii)\ A \cap (A \cup B) = (A \cap A) \cup (A \cap B) = A \cup (A \cap B) = A \quad [\text{(i)}]
Q8. Show that A ∩ B = A ∩ C need not imply B = C.दिखाइए कि A ∩ B = A ∩ C से B = C होना आवश्यक नहीं।Show that A ∩ B = A ∩ C need not imply B = C.
  1. Example: A = {1, 2}, B = {1, 3}, C = {1, 4}.उदाहरण: A = {1, 2}, B = {1, 3}, C = {1, 4}।Example: A = {1, 2}, B = {1, 3}, C = {1, 4}.
    A \cap B = \{1\} = A \cap C,\quad B \ne C
Q9. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X, show that A = B.यदि किसी समुच्चय X के लिए A ∩ X = B ∩ X = φ और A ∪ X = B ∪ X, तो दिखाइए कि A = B।If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X, show that A = B.
  1. A = A \cap (A \cup X) = A \cap (B \cup X) = (A \cap B) \cup (A \cap X) = A \cap B
  2. B = B \cap (B \cup X) = B \cap (A \cup X) = (B \cap A) \cup (B \cap X) = A \cap B

✔ So A = A \cap B = B.अतः A = A \cap B = B।Isliye A = A \cap B = B.

Q10. Find sets A, B, C such that A ∩ B, B ∩ C and A ∩ C are non-empty and A ∩ B ∩ C = φ.ऐसे समुच्चय A, B, C ज्ञात कीजिए कि A ∩ B, B ∩ C, A ∩ C अरिक्त हों और A ∩ B ∩ C = φ।Find sets A, B, C such that A ∩ B, B ∩ C and A ∩ C are non-empty and A ∩ B ∩ C = φ.
  1. Give each pair its own common element.हर युग्म को उसका अपना उभयनिष्ठ अवयव दीजिए।Har pair ko uska apna common element dijiye.
    A = \{1, 2\},\ B = \{2, 3\},\ C = \{1, 3\}:\ A \cap B = \{2\},\ B \cap C = \{3\},\ A \cap C = \{1\},\ A \cap B \cap C = \phi

🎯 Quiz — 12 questions🎯 क्विज़ — 12 प्रश्न🎯 Quiz — 12 questions

Tap an option. The answer and a short reason appear at once.किसी विकल्प पर टैप करें। उत्तर और छोटा कारण तुरंत दिखेगा।Kisi option par tap kijiye. Answer aur chhota reason turant dikhega.

1. Which of these is a set?इनमें से कौन समुच्चय है?Inme se kaun set hai?

2. The letters of the word ASSASSINATION in roster form have how many elements?ASSASSINATION शब्द के अक्षरों के समुच्चय में कितने अवयव हैं?ASSASSINATION ke letters ke set mein kitne elements hain?

3. How many subsets does {a, b, c, d} have?{a, b, c, d} के कितने उपसमुच्चय हैं?{a, b, c, d} ke kitne subsets hain?

4. {x ∈ R : −2 ≤ x < 5} as an interval is{x ∈ R : −2 ≤ x < 5} अंतराल रूप में है{x ∈ R : −2 ≤ x < 5} interval form mein hai

5. If A = {1, 2, 3} then which is correct?यदि A = {1, 2, 3} तो कौन-सा सही है?Agar A = {1, 2, 3} to kaun-sa sahi hai?

6. A = {2, 4, 6, 8}, B = {4, 8, 12}. A − B = ?A = {2, 4, 6, 8}, B = {4, 8, 12}। A − B = ?A = {2, 4, 6, 8}, B = {4, 8, 12}. A − B = ?

7. If A ⊂ B, then A ∩ B equalsयदि A ⊂ B, तो A ∩ B बराबर हैAgar A ⊂ B, to A ∩ B barabar hai

8. (A ∪ B)′ is equal to(A ∪ B)′ बराबर है(A ∪ B)′ barabar hai

9. Which set is empty?कौन-सा समुच्चय रिक्त है?Kaun-sa set empty hai?

10. In N, the complement of the set of prime numbers isN में अभाज्य संख्याओं के समुच्चय का पूरक हैN mein primes ke set ka complement hai

11. Which pair of sets is disjoint?कौन-सा युग्म असंयुक्त है?Kaun-sa pair disjoint hai?

12. A ∪ A′ = ?

← MathsCh 2 →

Found a mistake?कोई गलती दिखी?Koi galti dikhi? Report a mistakeगलती बताइएGalti bataiye — we fix confirmed errors quickly.— पक्की गलतियाँ हम जल्दी ठीक करते हैं।— confirm galtiyan hum jaldi theek karte hain.

These are Simple Learning Hub's own study notes and solutions, written to follow the NCERT Class 11 Mathematics textbook (Reprint 2026-27) section by section; read them with the official NCERT chapter. Diagrams are original SLH drawings.ये Simple Learning Hub के अपने स्टडी नोट्स हैं, जो NCERT कक्षा 11 गणित की किताब (2026-27) के हर अनुभाग के साथ लिखे गए हैं; इन्हें NCERT के आधिकारिक अध्याय के साथ पढ़िए। डायग्राम SLH के अपने बनाए चित्र हैं।Ye Simple Learning Hub ke apne study notes hain, jo NCERT Class 11 Maths ki kitaab (2026-27) ke har section ke saath likhe gaye hain; inhe NCERT ke official chapter ke saath padhiye. Diagrams SLH ki apni drawings hain.