Let Z be the set of integers. Show that the relation R = {(a, b):

1.	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.
Answer» Given as R = {(a, b): a, b ∈ Z and a + b is even} is a relation defined on R. Given Z be the set of integers To prove equivalence relation the relation should be reflexive, symmetric and transitive. We have to check these properties on 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 Thus, 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 Therefore, 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 above two equations, 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 Clearly, R is transitive on Z. ∴ R is reflexive, symmetric and transitive. Hence, R is an equivalence relation on Z

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.

Answer»

Given as R = {(a, b): a, b ∈ Z and a + b is even} is a relation defined on R.

Given Z be the set of integers

To prove equivalence relation the relation should be reflexive, symmetric and transitive.

We have to check these properties on 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

Thus, 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

Therefore, 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 above two equations, 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

Clearly, R is transitive on Z.

∴ R is reflexive, symmetric and transitive.

Hence, R is an equivalence relation on Z