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.
R = {(a, b) : a ≤ b}
(i) Since (a, a) ∈ R ∀ a ∈ R [∵ a ≤ a ∀ a ∈ R]
∴ R is reflexive.
(ii) (a, b) ∈ R ⇏ (b, a) ∈ R [∵ if a ≤ b. then b ≤ a is not true]
∴ R is not symmetric.
(iii) Let (a, b), (b, c) ∈ R ∴ a ≤ b, b ≤ c ∴ a ≤ c ⇒ (a, c) ∈ R ∴ (a, b), (b. c) ∈ R ⇒ (a, c) ∈ R ∴ R is 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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