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Proof of a demonstration in Discrete Math
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Prove that P(A)P(B) if and only if AB. [duplicate]up vote 8 down vote favorite This question already has an answer here: Prove/Disprove that if two sets have the same power set then they are the same set 3 answers Here is my proof, I would appreciate it if someone could critique it for me:To prove this statement true, we must proof that the two conditional statements ("If P(A)P(B), then AB," and, If AB, then P(A)P(B)) are true.
Contrapositive of the first statement: If AB, then P(A)P(B)If AB, then there must be some element in A, call it x, that is not in B: xA, and xB. Since xA, then {x}P(A); moreover, since xB, then {x}P(B), which proves that, if AB, then P(A)P(B). By proving the contrapositive true, the original proposition must be true.
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