Use element argument method to prove that for any two sets
A and B, if A ⊆ B, then P(A) ⊆ P(B), where P(A) and P(B) are power sets of A and B respectively. You must state your reasons clearly for every statement in your proof.
Proof.
Let A and B be sets, such that A ⊆ B.
Let X in P(A).
From definition of power set, X ⊆ A.
Let y in X.
.........................
........., y in B.
Thus, from definition of subset, we have X ⊆ B.
Hence, from the definition of power set, X in P(B).
Therefore, from the definition of subset, we have P(A) ⊆ P(B).