Let Z be the set of integers. Show that the relation R = {(a, b) : a, b ∈ Z and a + b is even} is an equivalence relation on Z.
We have,
Z = set of integers and
R = {(a, b) : a, b ∈ Z and a + b is even} be a relation on Z.
To prove: R is an equivalence relation on Z.
Proof :
To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.
Reflexivity : For Reflexivity, we need to prove that-
(a, a) ∈ R
Let a ∈ Z
⇒ a + a is even
⇒ (a, a) ∈ R
⇒ R is reflexive
Symmetric: For Symmetric, we need to prove that-
If (a, b) ∈ R, then (b, a) ∈ R
Let a, b ∈ Z and (a, b) ∈ R
⇒ a + b is even
⇒ b + a is even
⇒ (b, a) ∈ R
⇒ R is symmetric
Transitive : : For Transitivity, we need to prove that-
If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R
Let (a, b) ∈ R and (b, c) ∈ R For some a, b, c ∈ Z
⇒ a + b is even and b + c is even
[if b is odd, then a and c must be odd ⇒ a + c is even,
If b is even, then a and c must be even ⇒ a + c is even]
⇒ a + c is even
⇒ (a, c) ∈ R
⇒ R is transitive
Hence, R is an equivalence relation on Z