Question

Prove that the following relation on the coordinate plane is an equivalence relation:$\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{~}\mathbf{(}\mathbf{u}\mathbf{,}\mathbf{v}\mathbf{)}$ if and only if x-u is an integer.

Short answer

It is proved that given relation$~$is equivalence.

Step-by-step solution

Definitions of Equivalence relation

A relation R defined on a set is called an equivalence relation if it is:

Reflexive

Symmetric

Transitive.

Prove that the given relation is an equivalence relation 

For reflexive,

Let $(x,y)~(x,y)$as.$x-x=0\in \mathbb{Z}$ So,$~$ is reflexive.

For symmetric,

Let $(x,y)~(u,v)$then$x-u$ is an integer. Also, $-(x-u)$is an integer. That is$u-x$, is an integer which implies$(u,v)~(x,y)$. So, $~$is symmetric.

For transitive,

Let$(x,y)~(u,v)$ and$(u,v)~(l,m)$ then. To show$(x,y)~(l,m)$ which is possible only when$x-l$ is an integer. Since addition of two integers is an integer. Therefore,$x-u+u-l=x-l$ .

Since,$(x-l)$ is an integer. Thus, $(x,y)~(l,m)$. So, $~$is transitive.

Since all the conditions are satisfied.

Therefore, the relation$~$ is an equivalence relation.