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Question
Show that the relation R on the set Z of integers, given by
R = {(a, b) : 2 divides a – b}, is an equivalence relation.
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Solution
We observe the following properties of relation R.
Reflexivity :
Let a be an arbitrary element of the set Z. Then,
a ∈ R
⇒ a−a = 0 = 0 × 2
⇒ 2 divides a − a
⇒ ( a, a ) ∈ R for all a ∈ Z
So, R is reflexive on Z.
Symmetry:
Let (a, b)∈ R
⇒ 2 divides a−b
⇒ `(a-b)/2`=p for some p ∈ Z
⇒ `(b-a)/2 = - p `
Here, −p ∈ Z
⇒ 2 divides b − a
⇒ (b, a)∈ R for all a, b ∈ Z
So, R is symmetric on Z
Transitivity :
Let (a, b) and (b, c) ∈ R
⇒ 2 divides a−b and 2 divides b−c
⇒ `(a-b)/2` = p and` (b-c)/2`= q for some p, q ∈ Z`
Adding the above two, we get
`(a-b)/2 + (b -c)/2 = p +q`
⇒ `(a -c)/2 p +q`
Here, p+ q ∈ Z
⇒2 divides a − c
⇒ (a, c)∈ R for all a, c ∈ Z
So, R is transitive on Z.
Hence, R is an equivalence relation on Z.
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