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Ex 1.3, 10 - Chapter 1 Class 12 Relation and Functions
Last updated at Dec. 16, 2024 by Teachoo
Inverse of function: Proof questions
Chapter 1 Class 12 Relation and Functions
Concept wise
- Relations - Definition
- Empty and Universal Relation
- To prove relation reflexive, transitive, symmetric and equivalent
- Finding number of relations
- Function - Definition
- To prove one-one & onto (injective, surjective, bijective)
- Composite functions
- Composite functions and one-one onto
- Finding Inverse
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Inverse of function: Proof questions
- Binary Operations - Definition
- Whether binary commutative/associative or not
- Binary operations: Identity element
- Binary operations: Inverse
Transcript
Ex 1.3, 10 Let f: X → Y be an invertible function. Show that f has unique inverse. (Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y, fog1(y) = IY(y) = fog2(y). Use one-one ness of f). Let f: X → Y be an invertible function. If g is an inverse of f, then for all y ∈ Y fog(y) = IY Let g1 & g2 be two inverses of f Then, for all y ∈ Y, fog1(y) = IY & fog2(y) = IY ⇒ fog1(y) = fog2(y) f (g1(y)) = f (g2(y)) Since f is invertible, f is one-one. So, if f(x1) = f(x2), then x1 = x2 Since f (g1(y)) = f (g2(y)), ∴ g1(y) = g2(y) ⇒ g1 = g2 Hence, f has a unique inverse.
