Let g1 and g2 be two inverses of f for all y ∈ Y, we have, (f o g1)(y) = y = IY (y) and (f o g 2) (y) = y = IY (y)
⇒ (f o g1) (y) = (f o g2) (y) for all y ∈ Y
⇒ f(g1(y)) = f(g2(y)) for all y ∈ Y
⇒ g1 (y) = g2(y) [∵ f is one-one as f is invertible]
∴ g1 = g2 ∴ inverse of f is unique.
Determine whether each of the following relations are reflexive, symmetric and transitive :
(i) Relation R in the set A = {1, 2, 3,....., 13, 14} defined as
R = {(x, y) : 3 x – y = 0}
(ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x,y) : y is divisible by x} (iv) Relation R in the set Z of all integers defined as R = {(x,y) : x – y is an integer}
(v) Relation R in the set A of human beings in a town at a particular time given by
(a) R = {(x, y) : x and y work at the same place}
(b) R = {(x,y) : x and y live in the same locality}
(c) R = {(x, y) : x is exactly 7 cm taller than y}
(d) R = {(x, y) : x is wife of y}
(e) R = {(x,y) : x is father of y}
Mock Test Series
