Show that the relation \(R\) on the set \(Z\) of integers, given by \(R=\left\{ \left( a,b \right) :2\ \text{divides}\ a-b \right\} \) is an equivalence relation.

The relation \(R\) on \(Z\) is given by \(R=\left\{ \left( a,b \right) :2\quad \text{divides}\quad a-b \right\} \)

We observe the following properties of relation \(R\)

We observe the following properties of relation \(R\)

Reflexivity: for any \(a\in Z\)

\(a-a=0=0\times 2\)

\(\Rightarrow\) \(2\) divides \(a-a\Rightarrow (a,a)\in R\)

So, \(R\) is relexive relation on \(Z\)

\(a-a=0=0\times 2\)

\(\Rightarrow\) \(2\) divides \(a-a\Rightarrow (a,a)\in R\)

So, \(R\) is relexive relation on \(Z\)

Symmetry: Let \(a,b\in Z\) be such that

\((a,b)\in R\)

\(\Rightarrow\) \(2\) divides \(a-b\)

\(\Rightarrow\) \(a-b=2\lambda\) for some \(\lambda \in Z\)

\(\Rightarrow\) \(b-a=2(-\lambda)\), where \(-\lambda \in Z\)

\(\Rightarrow\) \(2\) divides \(b-a\Rightarrow (b,a)\in R\)

Thus \((a,b)\in R\Rightarrow (b,a)\in R\). So, \(R\) is a symmetric relation on \(Z\).

\((a,b)\in R\)

\(\Rightarrow\) \(2\) divides \(a-b\)

\(\Rightarrow\) \(a-b=2\lambda\) for some \(\lambda \in Z\)

\(\Rightarrow\) \(b-a=2(-\lambda)\), where \(-\lambda \in Z\)

\(\Rightarrow\) \(2\) divides \(b-a\Rightarrow (b,a)\in R\)

Thus \((a,b)\in R\Rightarrow (b,a)\in R\). So, \(R\) is a symmetric relation on \(Z\).

Transitivity. Let \(a,b,c\in Z\) be such that \((a,b)\in R\) and \((b,c)\in R\). Then,

\((a,b)\in R\Rightarrow 2\) divides \(a-b\Rightarrow a-b=2\lambda\) for some \(\lambda \in Z\)

and \((b,c)\in R\Rightarrow 2\) divides \(b-c\Rightarrow b-c=2\mu\) for some \(\mu \in Z\)

\(\therefore\) \(a-b+b-c=2(\lambda +\mu)\)

\(\Rightarrow\) \(a-c=2(\lambda +\mu)\), where \(\lambda +\mu \in Z\)

\(\Rightarrow\) \(2\) divides \(a-c\)

\(\Rightarrow\) \((a,c)\in R\)

thus, \((a,b)\in R\) and \((b,c)\in R\Rightarrow (a,c)\in R\)

So, \(R\) is a transitive relation on \(Z\)

Hence, \(R\) is an equivalence relation on \(Z\)

\((a,b)\in R\Rightarrow 2\) divides \(a-b\Rightarrow a-b=2\lambda\) for some \(\lambda \in Z\)

and \((b,c)\in R\Rightarrow 2\) divides \(b-c\Rightarrow b-c=2\mu\) for some \(\mu \in Z\)

\(\therefore\) \(a-b+b-c=2(\lambda +\mu)\)

\(\Rightarrow\) \(a-c=2(\lambda +\mu)\), where \(\lambda +\mu \in Z\)

\(\Rightarrow\) \(2\) divides \(a-c\)

\(\Rightarrow\) \((a,c)\in R\)

thus, \((a,b)\in R\) and \((b,c)\in R\Rightarrow (a,c)\in R\)

So, \(R\) is a transitive relation on \(Z\)

Hence, \(R\) is an equivalence relation on \(Z\)

To Keep Reading This Answer, Download the App

4.6

Review from Google Play

To Keep Reading This Answer, Download the App

4.6

Review from Google Play

Correct7

Incorrect0

Still Have Question?

Load More

More Solution Recommended For You