Short Answer TypeShow that the relation R in the set A of all the books in a library of a college given by R = {(x, y): x and y have same number of pages} is an equivalence relation.
Let A = {1, 2, 3, 4, 5, 6}
R = {(a, b) : b = a + 1} = {(a, a + 1)}
= {(1, 2), (2, 3), (3, 4), (4,5), (5,6)}
(i) R is not reflexive as (a, a) ∉ R ∀ a ∈ A
(ii) (a,b) ∈ R ⇏ (b,a) ∈ R [∵ (a, b) ∈ R ⇒ b = a + 1 ⇒ a = b –1]
∴ R is not symmetric.
(iii) (a, b) ∈ R, (b, c) ∈ R ⇏ (a, c) ∈ R
[∵ (a, b), (b, c) ∈ R ⇒ b = a + 1, c = b + 1 ⇒ c = a + 2]
∴ R is not transitive.
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}
Determine whether each of the following relations are reflexive, symmetric and transitive :
(ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}
Determine whether each of the following relations are reflexive, symmetric and transitive :
(ii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x,y) : y is divisible by x}
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