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 ≤ b3}
(i) Since (a, a) ∉ R as a ≤ a3 is not always true
[Take a = 1/3. then a ≤ a3 is not true]
∴ R is not reflexive.
(ii) Also (a, b) ∈ R ⇏ (b, a) ∈ R
[Take a = 1, b = 4, ∴ 1 ≤ 43 but 4 ≰ (l)3 ]
∴ R is not symmetric.
(iii) Now (a, b) ∈ R, (b, c) ∈ R ⇏ (a, c) ∴ R
[Take a = 100, b = 5, c = 3, ∴ 100 ≤ 53, 5 ≤ 33 but 100 ≥ 33] R is not symmetric.
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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