One-one, many-one and onto: four arrow diagramsएकैकी, बहुएक और आच्छादक: चार तीर-आरेखOne-one, many-one aur onto: chaar arrow diagrams

NCERT Fig 1.2 · Class 12 Maths · Chapter 1 · interactive diagram

Parts and what they mean

• Function f₁ (फलन f₁) — Different elements go to different images (1→a, 2→b, 3→d, 4→c): one-one. But e and f get no arrow, so f₁ is not onto.
• Function f₂ (फलन f₂) — 1 and 2 both go to b, so f₂ is many-one; a, e, f get no arrow, so it is not onto either.
• Function f₃ (फलन f₃) — 1 and 2 both go to a (many-one), but every element a, b, c of X₃ gets an arrow, so f₃ is onto.
• Function f₄ (फलन f₄) — Distinct elements have distinct images AND every element of X₄ is hit: one-one and onto (bijective).
• e — not an image (e — किसी का प्रतिबिंब नहीं) — No element of X₁ goes to e, so Range of f₁ ≠ X₂.
• f — not an image (f — किसी का प्रतिबिंब नहीं) — No element of X₁ goes to f either.
• b — image of two elements (b — दो अवयवों का प्रतिबिंब) — f₂(1) = f₂(2) = b with 1 ≠ 2: this is exactly what "many-one" means.
• a — image of 1 and 2 (a — 1 और 2 का प्रतिबिंब) — Two arrows land on a, so f₃ is not one-one.
• Domain X₁ = {1, 2, 3, 4} (प्रांत X₁ = {1, 2, 3, 4}) — Every element of the domain sends exactly one arrow (that makes it a function).
• Co-domain X₂ (सहप्रांत X₂) — The set the arrows go into. If every element here is hit, the function is onto.
• (i) one-one, not onto ((i) एकैकी, आच्छादक नहीं) — The type of this function.
• Domain X₁ = {1, 2, 3, 4} (प्रांत X₁ = {1, 2, 3, 4}) — Every element of the domain sends exactly one arrow (that makes it a function).
• Co-domain X₂ (सहप्रांत X₂) — The set the arrows go into. If every element here is hit, the function is onto.
• (ii) many-one, not onto ((ii) बहुएक, आच्छादक नहीं) — The type of this function.
• Domain X₁ = {1, 2, 3, 4} (प्रांत X₁ = {1, 2, 3, 4}) — Every element of the domain sends exactly one arrow (that makes it a function).
• Co-domain X₃ (सहप्रांत X₃) — The set the arrows go into. If every element here is hit, the function is onto.
• (iii) many-one, onto ((iii) बहुएक, आच्छादक) — The type of this function.
• Domain X₁ = {1, 2, 3, 4} (प्रांत X₁ = {1, 2, 3, 4}) — Every element of the domain sends exactly one arrow (that makes it a function).
• Co-domain X₄ (सहप्रांत X₄) — The set the arrows go into. If every element here is hit, the function is onto.
• (iv) one-one and onto ((iv) एकैकी तथा आच्छादक) — The type of this function.

Original diagram drawn by Simple Learning Hub, based on NCERT Fig 1.2. Report a mistake