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Q2SE

Expert-verifiedFound in: Page 187

Book edition
7th

Author(s)
Kenneth H. Rosen

Pages
808 pages

ISBN
9780073383095

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

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

$\overline{)O}$ 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" $X\subseteq Y$

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 $\overline{)O}$,the proof is trivial as any power set contains the empty set .

If $\rho \left(A\right)$ does not contain only the empty set $\overline{)O}$, 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 $\rho \left(B\right)$.

$\rho \left(A\right)\u2291\rho \left(B\right)$

Hence the solution is,

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

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