Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

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

$ρ (A) \subseteq ρ (B)$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1:

$O$ 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 : $ρ (A) \subseteq ρ (B)$

PROOF:

If role="math" localid="1668425925367" $ρ (A)$ contains only the empty set $O$ ,the proof is trivial as any power set contains the empty set .

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

$\{x\} ε ρ (A)$

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.

$x ε A$

Since $A \subseteq B$

$x ε B$

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 $ρ (A)$ also has to be in $ρ (B)$ .

By the definition of a subset, we know that $ρ (A)$ is a subset of $ρ (B)$ .

$ρ (A) ⊑ ρ (B)$

Hence the solution is,

$ρ (A) \subseteq ρ (B)$

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Discrete Mathematics and its Applications

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