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}

A is the set of human beings in a town at a particular time R is relation in A.
(a) R = {(x, y) : x and y work at the same time}
R is reflexive as (x, x) ∈ R R is symmetric as ( x, y) ∈ R ∈ (y, x) ∈ R
[ ∵ x and y work at the same time ⇒ y and x work at the same time] R is transitive as (x, y), (y, z) ∈ R ⇒ (x, z) ∈ R
[∴ if x and y, y and z work at the same time, then x and z also work at the same time]
(b) R = {(x,y) : x and y live in the same locality}
R is reflexive as (x, x) ∈ R R is symmetric as ( x, y) ∈ R ⇒ (y, x) ∈ R
[∴ x and y live in the same locality ⇒ y and x live in the same locality] R is transitive as ( x, y), ( y, z) ∈ R ⇒ (x, z) ∈ R
[∵ if x and y, y and z live in the same locality. then x and z also live in the same locality]
(c) R = {(x,y) : x is exactly 7 cm taller than y}
Since (x, x) ∉ R as x cannot be 7 cm taller than x.
∴ R is not reflexive.
(x, y) ∈ R ⇒ (y.x) ∈ R as if x is taller than y, then y cannot be taller than x.
∴ R is not symmetric.
Again (x,y), (y,z) ∈ R ⇏ (x, z) ∈ R
[∵ if x is taller than y by 7 cm and y is taller than z by 7 cm,
then x is taller than z by 14 cm]
∴ R is not transitive.
(d) R = {(x,y) : x is wife of y}
R is not reflexive as (x,y) ∉ R    [∴ x cannot be wife of x]
Also (x, y) ∈ R ⇏ (y, x) ∈ R [∵ if x is wife of y, then y cannot be wife of x] ∴ R is not symmetric.
R is not transitive.
(e) R = {(x,y) : x is father of y}
R is not reflexive as (x, x) ∉ R    [ ∵ x cannot be father of x]
Also (x,y) ∈ R ⇐ (y, x) ∈ R [ ∵ if x is father of y. then y cannot be father of x] ∴ R is not symmetric.
R is not transitive.