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Question
Show that the relation R on the set Z of all integers, given by R = {(a,b) : 2 divides (a-b)} is an equivalence relation.
Solution
To prove relation is an equivalence relation
We have to show three properties
1. Reflexive
(a,a) ∈ R
2. Symmetric
(a,b) ∈ R
⇒ (b,a) ∈ R
3. Transitive
(a,b) ∈ R and (b,c) ∈ R
⇒ (a,c) ∈ R
1. R is reflexive because 2 divides (a - a)∀a ∈ z∀a ∈ z
2. 2 divides a - b
therefore, 2 divides b - a hence, (b,a) ∈ R
R is symmetric
3. (a, b) ∈ R
(b, c) ∈ R
then a − b and b − c are divisible by 2.
Now, a − c = ( a − b ) + ( b − c) = a−c
so, a − c is divisible by 2.
Therefore, (a, c ) ∈ R
Therefore, R is an equivalence relation.
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