If $A$ is a set, then suppose that $f$ is a one-to-one function from $A$ to $P(A)$, the power set of $A$ and let $B = \{a \in A | a \notin f(a)\}$. For the following sets, give examples of at least three different functions from $A$ to $P(A)$ and construct the set $B$:
(a) $A = \{-1,0,1\}$
(b) $A = \{1,2,3,\dots\}$
I'm confused by the idea of a function from a set to its power set. Can I just say, for example:
$f:A \rightarrow P(A)$, $\{-1,\{-1\}\}$ making $B=\{0,1\}$?
Best Answer
A function from $A$ to $P(A)$ is just like any other function: in this case the domain of the function is $A$, and its range is all subsets of $A$. So for example one function from $A=\{-1,0,1\}$ to $P(A)$ might be $$-1 \mapsto \{-1\},\quad 0\mapsto \{0\},\quad 1\mapsto \{1\},$$ and for this function $B$ would be empty, since in each case $a\in f(a)$. Another example might be $$-1\mapsto \{-1, 0\},\quad 0\mapsto \emptyset,\quad 1\mapsto \{-1\},$$ and for this function $B = \{0, 1\}$, since for these elements of $A$, we have $a\notin f(a)$. Can you take it from here?