Suppose A, B are sets. Prove: If A ⊆ B, then P(A) ⊆ P(B).
Note P(A)= the power set of A and P(B)= the power set of B.
Please go step by step because i dont understand where to go or even how to do this problem and state what kind of proof you are using (ie. direct, counterexample, contrapositive, basic induction, etc.)
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Direct Proof: Assume that A ⊆ B, and derive that P(A) ⊆ P(B).
Let S ∈ P(A).
Then by the definition of the power set, S ⊆ A.
But since A ⊆ B, we know by the transitivity of subsets that S ⊆ B.
So by the definition of the power set, we have S ∈ P(B).
Thus, since we have shown that S ∈ P(A) implies that S ∈ P(B),
it follows by the definition of subset that P(A) ⊆ P(B), as desired.