Question

In the set Z of integers define mRn if m-n is multiple of 12 prove that R is an equivalence relation​

Answer 1

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 $aRa, \forall a\in Z$

2. Symmetric

if $aRb\Rightarrow bRA \ \forall a,b\in Z$

3. Transitive:

$aRb, bRc \Rightarrow aRc \ \forall a,b,c \in Z$

The given relation is:

$R = \{(m,n) : m-n \text{ is a multiple of 12}; m,n\in Z\}$

1. Reflexive:

$mRm = m-m=0$

0 is a multiple of 12, $\therefore \bold{reflexive}$.

2. Symmetric:

$mRn\Rightarrow m-n=12a$

$nRM\Rightarrow n-m=-12a$

Both are multiple of 12, $\therefore \bold{symmetric}$.

3. Transitive:

$mRn\Rightarrow m-n=12a$ .... (1)

$nRo\Rightarrow n-o=12b$ .... (2)

Adding (1) and (2):

$m-o=12(a+b)$

Hence, $mRo$. $\therefore \bold{transitive}$.

Hence proved that the relation R is equivalence relation.