Math 2.0

definition

Power set

Definition

Let AA be a set. The power set of AA is the set P(A)={S∣S⊆A}\mathcal{P}(A) = \{S \mid S \subseteq A\} of all subsets of AA.

Remarks

Other notations are 2A2^A and ℘(A)\wp(A). Example: P({1,2})={∅,{1},{2},{1,2}}\mathcal{P}(\{1, 2\}) = \{\varnothing, \{1\}, \{2\}, \{1, 2\}\}. A finite set with nn elements has a power set with 2n2^n elements. See cardinality of the power set. By Cantor's theorem, no set is in bijection with its power set.

Explanations

L1 human

To list the power set, choose for each element whether it is in or out of the subset. Each choice gives one subset, so a set with 33 elements has 23=82^3 = 8 subsets. The empty set and the set itself are always among them.

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  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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Sources

  1. Web https://en.wikipedia.org/wiki/Power_set, lead section

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