☰
Relations and Functions
Let R and R’ be two symmetric relations on a set A.
Let a, b ∈ A such that (a, b) ∈ R ∪ R’
∴ Either (a, b) ∈ R or (a, b) ∈ R’
If (a, b) ∴ R then (b, a) ∴ R (∵ R is symmetric)
∴ (b, a) ∈ R ∪ R’ (since R ⊆ R⊆ R’)
Similarly we can prove that (a, b) ∈ R’ ∈ (b, a) ∈ R ∪ R’
In both the cases (b, a) ∈ R ∪ R’
∴ R ∪ R’ is a symmetric relation on A.
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