Proof: A=B iff P(A)=P(B) (Sets are Equal iff their Power Sets are Equal) | Set Theory

Let A and B be sets. Then A=B if and only if P(A)=P(B). That is, two sets are equal if and only if their power sets are equal. We prove this basic set theory result in today's lesson.

First, we want to prove that if A equals B then P(A) equals P(B). First, take an element S from P(A). Thus, S is a subset of A. But A=B, so S is a subset of B. Thus, S is an element of P(B). Hence, P(A) is a subset of P(B). The same logic works to prove P(B) is a subset of P(A), and so P(A)=P(B). Note that a power set is never empty, so we don't have to make any special mention of the empty set here.

Next, we want to prove that if P(A)=P(B) then A=B. Take an element, x, from A (if A is empty, then A is a subset of B trivially). Then {x} is a subset of A and so {x} is an element of P(A). But P(A)=P(B), so {x} is an element of P(B), thus {x} is a subset of B and so x is in B. Thus, A is a subset of B. The same logic shows B is a subset of A and so A=B.

Proof A is a Subset of B iff B' is Subset of A':    • Proof: A is a Subset of B iff B' is S...  

#settheory #math

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