Question

Question: Let$\mathit{f}\mathbf{:}\mathit{A}\mathbf{\to }\mathit{B}$be a function. Prove that the following relation$\mathit{A}\mathbf{:}\mathit{u}\mathbf{~}\mathit{v}$is an equivalence relation of if and only if $\mathit{f}\left(u\right)\mathbf{=}\mathit{f}\left(v\right)$.

Short answer

Answer:

It is proved that given relation$A:u~v$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 

$v~u$For reflexive,

Let $u~uasf\left(u\right)=f\left(u\right)$. So, is reflexive.

For symmetric,

Let$u~v$then $f\left(u\right)=f\left(v\right)$implies$f\left(v\right)=f\left(u\right)$that is.$v~u$So, implies. So, $u~v$is symmetric .

For transitive,

Let $u~v$and $v~w$then $f\left(u\right)=f\left(v\right)$and $f\left(v\right)=f\left(w\right)$. To show, $u~v$for this need to show$f\left(u\right)=f\left(w\right)$.

Since,

$f\left(u\right)=f\left(v\right)$

And,

$f\left(v\right)=f\left(w\right)$

On equating both$f\left(u\right)=f\left(w\right)$

Thus, $f\left(u\right)=f\left(w\right)$.

Hence,$u~v$ and$v~w$ implies that $u~w$. So, $~$is transitive.

Since all the conditions are satisfied. Therefore, the relation is an equivalence relation.