the relation R in X is empty relation if
(a) R = null is a subset of X x X
(b) R = X+X
also explain the answer
Answers
Answer:
RELATIONS AND FUNCTIONS 7
This leads to a function g : R → R defined as
g (y) =
5
4
y −
. Therefore, (g o f ) (x) = g(f (x) = g (4x + 5) = 4 5 5
4
x+ −
= x
or g o f = IR Similarly (f o g) (y) = f (g(y)) = 5
4
y
f −
= 5
4 5
4
y −
+
= y
or f o g = IR
. Hence f is invertible and f –1 = g which is given by
f –1 (x) =
5
4
x −
Example 16 Let *
be a binary operation defined on Q. Find which of the following
binary operations are associative
(i) a * b = a – b for a, b ∈ Q.
(ii) a * b = 4
ab
for a, b ∈ Q.
(iii) a * b = a – b + ab for a, b ∈ Q. (iv) a * b = ab2
for a, b ∈ Q. Solution
(i) * is not associative for if we take a = 1, b = 2 and c = 3, then
(a * b) * c = (1 * 2) * 3 = (1 – 2) * 3 = – 1 – 3 = – 4 and
a * (b * c) = 1 * (2 * 3) = 1 * (2 – 3) = 1 – ( – 1) = 2.
Answer:
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