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The Power Set Cardinal Number
publish date: 2026/05/23 21:45:36.618870 UTC
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If a set \(X\) has cardinal number \(|X|\), what is the cardinal number of its power set \(P(X)\)?
Correct Answer
X
Explanation
The cardinal number of \(P(X)\) is \(2^{|X|}\). For finite sets: if \(|A| = n\), then \(|P(A)| = 2^n\) (e.g., \(n=3 \Rightarrow 2^3 = 8\) subsets). For infinite sets: \(|P(\mathbb{N})| = 2^{\aleph_0} = c\), and \(|P(\mathbb{R})| = 2^c > c\). A key axiom states: the cardinal of any nonempty set \(X\) is strictly less than \(2^{|X|}\), producing an endless hierarchy of infinities.
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012