Question

Suppose that A and B are sets such that the power set of A is a subset of the power set of power set of B. does it follow that A is a subset of B?

Short answer

A is a subset of B : Yes

Step-by-step solution

Step 1:

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:$X\subseteq Y$

Step 2:

Given:

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

To proof : $A\subseteq B$

PROOF:

Let us assume $\rho \left(A\right)\subseteq \rho \left(B\right)$

Let x be an element of A.

$\left\{x\right\}\epsilon A$

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

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

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

$x\epsilon \rho \left(B\right)$

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.

$x\epsilon B$

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 $\rho \left(B\right)$.

$A\subseteq B$

Hence the solution is,

Yes