Question

In the set z of integers defined mRn if m-n is a multiple of 12. Prove that R is an equivalence relation

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Given:

Relation mRn defined on set of integers Z

m-n is a multiple of 12.

To prove:

The relation R is an equivalence relation.

Solution:

First of all, let us have a look at the definition of equivalence relation.

A relation R defined on set Z is known as equivalence relation if it is:

1. Reflexive:

if

2. Symmetric

if

3. Transitive:

The given relation is:

1. Reflexive:

0 is a multiple of 12, .

2. Symmetric:

Both are multiple of 12, .

3. Transitive:

.... (1)

.... (2)

Adding (1) and (2):

Hence, . .

Hence proved that the relation R is equivalence relation.