Determine whether each of the following relations are reflexive, symmetric and transitive:
(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}

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Solution

(a)R is reflexive, symmetric and transitive obviously.( For reflexive (x, x ) work in same place. For symmetric (x,y) and (y,x) work in same place. For transitive (x,y) , (y,z) work at same place place means (x,z) work at same place)
(b)R is reflexive, symmetric and transitive obviously.( For reflexive (x, x ) live in same locality. For symmetric (x,y) and (y,x) live in same locality. For transitive (x,y) , (y,z) live in same locality means (x,z) live in same locality)
(c)Here, R is not reflexive as x is not 7 cm taller than x itself . R is not symmetric as if x is exactly 7 cm taller than y, then y exactly 7 cm taller than x is incorrect and not transitive as x is exactly 7 cm taller than y and y is exactly 7 cm taller than z , then x is exactly 7 cm taller than z is incorrect
(d)Here, R is not reflexive; as x is not wife of x, R is not symmetric, as if x is wife of y. then y is husband (not wife)of x and R is not transitive as x is wife of y and y is wife of z is contradicted .
(e)Here, R is not reflexive; as x cannot be father of x. For any x, R is not symmetric as if x is father of y, then y cannot be father of x. R is not transitive as if x is father of y and y is father of z, then x is grandfather (not father)of z.

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Determine whether each of the following relations are reflexive, symmetric and transitive:

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(a) R = {(x, y): x and y work at the same place}

(b) R = {(x, y): x and y live in the same locality}

(d) R = {(x, y): x is wife of y}

(e) R = {(x, y): x is father of y}

Determine whether each of the following relations are reflexive, symmetric and transitive:

(iv) Relation R in the set Z of all integers defined as R = {(x, y): x − y is an integer}

Which one of the following is not a function ?

(a) ${(x, y) : x, y ϵ R, x^{2} = y}$
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