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Consider the set $A =\{3,4\}$. Its power set is $P(A) = \{\varnothing, \{3\}, \{4\}, \{3,4\}\}$. $2^A$ is defined as the set of all functions from $A$ to $2$. Letting $2 = \{\varnothing, \{\varnothing\}\}$, we have the cartesian product $$A \times 2 = \{\langle 3, \varnothing \rangle, \langle 3, \{\varnothing\}\rangle,\langle4,\varnothing\rangle,\langle4,\{\varnothing\}\rangle\}$$ Finally, we know that $f \in 2^A \iff f \in P(A \times 2)$ and $f$ is a function.

Notice that the power set $P(A\times2)$ will have the singleton of each element in $A\times2$ (which will be functions in the set-theoretical sense), and other combinations like $\{\langle3,\varnothing\rangle, \langle4,\varnothing\rangle\}$. But this will mean that there are more than 4 elements of $P(A \times 2)$ that are functions (in the set-theoretical sense of a relation), and that therefore $P(A)$ and $2^A$ aren't equinumerous.

Clearly I'm doing something very wrong here, would anyone have a hint?

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  • $\begingroup$ Your description of $f\in2^A$ is incorrect. you need $f$ to be a set-theoretic function and have domain equal to $A$. $\endgroup$ Aug 16, 2021 at 4:38
  • $\begingroup$ The bijection is trivial: given a function $f\colon A\to 2$, define the subset $A_f=\{a\in A\mid f(a)=1\}$. The inverse of this correspondence maps a subset $B\subseteq A$ to its indicator function $\xi_B\colon A\to 2$, $\xi_B(x) = 0$ if $x\notin B$, and $\xi_B(x)=1$ if $x\in B$. $\endgroup$ Aug 16, 2021 at 4:39
  • $\begingroup$ ah, that's what I was missing! thank you so much for the tip! $\endgroup$
    – shintuku
    Aug 16, 2021 at 4:40

1 Answer 1

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The $4$ members of $2^A$ are $\{(3,0)(4,0)\},\{(3,0)(4,1)\}\{(3,1)(4,0)\}\{(3,1)(4,1)\}.$ These are subsets of $A\times 2$ but there are $12$ other subsets of $A\times 2$. A member of $P(A\times 2)$ is a member of $2^A$ iff it is the characteristic (indicator) function of a subset of $A.$

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