Question

Show that if A is a subset of B, then the power set of A is a subset of the power set of B.

Short answer

$\rho \left(A\right)\subseteq \rho \left(B\right)$

Step-by-step solution

Step 1:

$\cancel{O}$ represents the empty set and the empty set does not contain any elemnets

The power set of S is the set of all subsets of S.

Notation: P(S)

X is a subset of Y if every element of X is also an element of Y.

Notation:role="math" localid="1668425827447" $X\subseteq Y$

Step 2:

Given:

$X\subseteq Y$

To proof :$\rho \left(A\right)\subseteq \rho \left(B\right)$

PROOF:

If role="math" localid="1668425925367" $\rho \left(A\right)$contains only the empty set $\cancel{O}$,the proof is trivial as any power set contains the empty set .

If$\rho \left(A\right)$ does not contain only the empty set$\cancel{O}$, then there exists a set of the form$\left\{X\right\}$ in$\rho \left(A\right)$.

$\left\{x\right\}\epsilon \rho \left(A\right)$

If the set containing only an element x is a set in the power set of , then the element x has to be an element of the set S.

$x\epsilon A$

Since$A\subseteq B$

$x\epsilon B$

If x is an element in a set S, then the set containing only that element x is a set in the power set of S.

We thus have derived that every element x in$\rho \left(A\right)$ also has to be in$\rho \left(B\right)$.

By the definition of a subset, we know that $\rho \left(A\right)$is a subset of $\rho \left(B\right)$.

$\rho \left(A\right)⊑\rho \left(B\right)$

Hence the solution is,

$\rho \left(A\right)\subseteq \rho \left(B\right)$