relation $R$ on power set $P(A)$

Question

I pretty confused about how to deal with this. how can I determine for a relation ($R$) on a power set whether it is reflective, symmetric, transitive. When $A$ is the final amount and $P(A)$ is the power set. Consider the relation $R$ of $P(A) $ given by:

$sRt \iff |s| = |t|$

I know that a relation is considered an equivalence relation if it satisfies reflexive, symmetric and transitive properties but Im having trouble to figure out how to show that in my example.

Answer 1

In general if $f$ is a function that has set $X$ as domain then the relation $\sim$ on $X$ that is prescribed by:$$x\sim y\iff f(x)=f(y)$$is an equivalence relation.

To verify that we only need to check that the following conditions are satisfied:

• $f(x)=f(x)$ for every $x\in X$ (reflexivity)
• $f(x)=f(y)\implies f(y)=f(x)$ for all $x,y\in X$ (symmetry)
• $f(x)=f(y)\text{ and }f(y)=f(z)\implies f(x)=f(z)$ (transitivity)

It is evident that these conditions are satisfied.

Now realize that in your case we are dealing with a function $|\cdot|$ that has powerset $\wp(A)$ as domain and that the relation $R$ is prescribed by:$$sRt\iff|s|=|t|$$

So what is said above immediately applies to your case.

Answer 2

Given $A$ is nonempty set and assume $\mid A \mid=n$, then $\mid \mathcal{P}(A)\mid=2^n$. Take $p,q,r\in\mathcal{P}(A)$. We'll prove the relation is equivalence relation.

1. Refleksive $pRq\iff \mid p\mid=\mid q\mid \iff \mid q\mid=\mid p\mid\iff qRp$
2. Symmetric $pRq\iff \mid p\mid=\mid q\mid\Longrightarrow \mid q \mid=\mid p \mid\iff qRp$
3. Transitive $pRq\iff \mid p\mid=\mid q\mid$, $qRr\iff \mid q\mid=\mid r\mid$, hence we get $$\mid p\mid=\mid q\mid=\mid r\mid \Longrightarrow \mid p\mid=\mid r\mid\iff pRr$$