Discrete Mathematics and Its Applications, Seventh Edition

Published by McGraw-Hill Education
ISBN 10: 0073383090
ISBN 13: 978-0-07338-309-5

Chapter 2 - Section 2.1 - Sets - Exercises - Page 126: 23

Answer

a) 8 b) 16 c) 2

Work Step by Step

Recall the power set of an n element set has $2^n$ elements. a) $\{ a, b, \{a,b\}\}$ has three elements, so $\mathcal{P}(\{a,b,\{a,b\}\})=2^3 = 8$. b) $\{ \emptyset , a, \{a\}, \{\{a\}\}\}$ has four elements, so $\mathcal{P}(\{ \emptyset , a, \{a\}, \{\{a\}\}\}) = 2^4 = 16$. c) We know $\mathcal{P}(\emptyset ) = \{\emptyset \}$, which is a one element set. So $\mathcal{P}(\mathcal{P}(\emptyset )) = \mathcal{P}(\{ \emptyset \}) = 2^1 = 2$.
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