☰
Relations and Functions
Here f : A → B and g : B → C are one-to-one functions ∴ g o f is a function from A to C.
Let x1, x2 ∈ A
Now (g o f) (x1) = (g o f) (x2)
⇒ g (f (x1)) = g (f(x2))
⇒ f(x1) = f(x2) [∵ g is one-to-one]
⇒ x1 = x2 [∵ f is one-to-one]
⇒ (g o f) (x1) = (g o f) (x2) ⇒ x1 = x2 , ∀ x1,x2 x1,x2 ∈ A
∴ gof is one-to-one.
Sponsor Area
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}
Sponsor Area
Sponsor Area