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Question
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.
Solution
We observe the following properties of R.
Reflexivity :
Let a be an arbitrary element of Z. Then,
a ∈ R
Clearly, a+a = 2a is even for all a ∈ Z.
⇒ (a, a) ∈ R for all a ∈ Z
So, R is reflexive on Z.
Symmetry :
Let (a, b) ∈ R
⇒ a+b is even
⇒ b+a is even
⇒ (b, a) ∈ R for all a, b ∈ Z
So, R is symmetric on Z.
Transitivity :
Let (a, b) and (b, c) ∈ R
⇒ a+b and b+c are even
Now, let a+b = 2x for some x ∈ Z
and b+c = 2y for some y ∈ Z
Adding the above two, we get
a+2b +c = 2x + 2y
⇒ a+c = 2 (x+y−b), which is even for all x, y, b ∈ Z
Thus, (a, c) ∈ R
So, R is transitive on Z.
Hence, R is an equivalence relation on Z
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