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.
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}
A = {1, 2, 3, 4, 5, 6}
R = {(x, y) : y is divisible by x}
(a) R is reflexive as (x, x) ∈ R ∀ x ∈ A [∴ x divides x ∀ x ∈ A]
(b) R is not symmetric as (1, 6) ∈ R but (6, 1) ∉ R.
(c) Let (x, y), (y, z) ∈ A
∴ y is divisible by x and z is divisible by y ∴ z is divisible by x
∴ (r, y) ∈ R (y, z) ∈ R ⇒ (x, z) ∈ R ∴ R is transitive.
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