Abstract Algebra: Theory and Applications · Problem 2.4.12

Groups — Thomas Judson

Mathematics — Abstract Algebra · power-sets-induction

Problem

Power Sets. Let XX be a set. Define the power set of XX, denoted P(X)\mathcal{P}(X), to be the set of all subsets of XX. For example, P({a,b})={∅,{a},{b},{a,b}}.\mathcal{P}(\{a,b\}) = \{\emptyset, \{a\}, \{b\}, \{a,b\}\}. For every positive integer nn, show that a set with exactly nn elements has a power set with exactly 2n2^n elements.

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