Show that P(A) is a subset of P(B) if A is a subset of B

September 18, 2009

Proof: Suppose A is a subset of B. Let x be any element of $\mathcal{P}(A)$, the power set of A. Then x is a subset of A by the definition of the power set. But because A is a subset of B, x must also be a subset of B. Hence x is an element of $\mathcal{P}(B)$ by the definition of power set. Because this condition holds for any element of $\mathcal{P}(A)$, $\mathcal{P}(A)$ is a subset of $\mathcal{P}(B)$. Q.E.D.

Does this proof work for uncountably infinite sets A and B? It is intuitively clear that this proof works for any finite set A. But if A was countably infinite, or uncountably infinite, then I’m not sure if this argument makes any sense at all.

• Mathematics,
• Set Theory,
• Proof