Define a relation

Let A={a,b,c} and P(A) be the power set of A ( the family of subsets of A)

Define the relation on P(A) defined by , if and only if B and C contain the same number of elements.

What is the question? (Hi)

he wants us to find the the relation of this power set?

Maybe you mean he wants you to explicitly LIST the members of the relation.

Let A = {a b c}

PA = {0 {a} {b} {c} {a b} {a c} {b c} {a b c}}

Let R = {<x y> | x in PA, y in PA, and x and y are equinumerous}

So R =

{<0 0>
<{a} {a}>
<{b} {b}>
<{c} {c}>
<{a} {b}>
<{b} {a}>
<{a} {c}>
<{c} {a}>
<{b} {c}>
<{c} {b}>
<{a b} {a b}>
<{a c} {a c}>
<{b c} {b c}>
<{a b} {a c}>
<{a c} {a b}>
<{a b} {b c}>
<{b c} {a b}>
<{a c} {b c}>
<{b c} {a c}>
<{a b c} {a b c}>}

That was boring. I have no idea why I bothered doing it.