Show that the relation R on the set of natural numbers defined as R: {

1.	Show that the relation R on the set of natural numbers defined as R: { (x, y): y – x is a multiple of 2} is an equivalance relation.
Answer» Since x – x = 0 is multiple of 2, (x, x) ∈ R Therefore reflexive. If y – x is a multiple of 2 then x – y is also a multiple of 2. Therefore (x, y) ∈ R ⇒ (y, x) ∈ R. Hence symmetric. If y – x is a multiple of 2 and z-y is a multiple of 2, then their sum y – x + z – y = z – x is a multiple of 2. Therefore (x, y), (y, z) ∈ R ⇒ (x, z) ∈ R Hence transitive. Therefore R is an equivalance relation.

Show that the relation R on the set of natural numbers defined as R: { (x, y): y – x is a multiple of 2} is an equivalance relation.

Answer»

Since x – x = 0 is multiple of 2, (x, x) ∈ R

Therefore reflexive.

If y – x is a multiple of 2 then x – y is also a multiple of 2. Therefore (x, y) ∈ R ⇒ (y, x) ∈ R. Hence symmetric.

If y – x is a multiple of 2 and z-y is a multiple of 2, then their sum y – x + z – y = z – x is a multiple of 2.

Therefore (x, y), (y, z) ∈ R ⇒ (x, z) ∈ R

Hence transitive.

Therefore R is an equivalance relation.