For each of the following relations on the set Z of interger

For each of the following relations on the set Z of intergers, determine if it is reflexive, symmetric, anti-symmetric or transitive. On the basis of these properties, state whether or not it is an equivalence relation or a partial order.

a) R = {(a,b)| a^2 = b^2}
b) S = {(a,b)| | a - b | <= 1}

Solution

(a)

R = {(a,b)| a^2 = b^2}

reflexive : (a,a) and (b,b) are satsifying the the reletion

symmetric: (a,b) => a^2=b^2

                    (b,a) => b^2 = a^2

so it is atsifying the the reletion

transitive: (a,b) => a^2=b^2

                    (b,c) => b^2 = c^2

from these 2

a^2 = c^2 => (a,c)

it is also atsifying the the reletion

so it is an equivalence relation

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(2)

R = {(a,b)| |a-b|<=1}

reflexive : (a,a) and (b,b) are satsifying the the reletion

symmetric: (a,b) => |a -b|<=1

                    (b,a) => |b-a|<=1

                          => |a -b |<=1

so it is atsifying the the reletion

transitive: (a,b) => |a -b|<=1

                    (b,c) => |b-c|<=1

from these 2 we can not say about

|a -c|<=1

it is not atsifying the the reletion

so it is a partial order.