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I am having some confusion in understanding, what exactly does a function from a set to a power set means. I don't want a proof to the cantor's theorem.

Consider a set $A = \{1,2,3\}$ , $P(A) = \{\{1,2\},\{2,3\}, \{1,3\},\ldots \}$ then what does the image $2 \to \{2,3\}$ mean? similarly $1 \to \{3\}$ or $ 3 \to \{5,6,7\}$ ?

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5 Answers

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Probably visualizing this will help:

This isn't really a function between A and P(A), since 3 doesn't land in P(A).
An example of an actual function:
This particular function sends each movie of the set "movies" to the year it came out.
You don't have to have a meaningful rule to build a function though; the only rules are:

• There's exactly one arrow starting from each element of the first set, and no other arrows
• Each arrow lands in an element of the second set.

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A function in set theory (and in most of mathematics in general) is just a set of ordered pairs $(a,b)$, such that no two pairs $(a,b)$, $(a,b')$ occur where $b\neq b'$. The idea is that the function associates the element $b$ to the element $a$.

For example, the function $f(x) = x^2$ on the real numbers is the set $\{(x,x^2)\;\mid\; x\in\mathbb{R}\}$.

So a function from a set $S$ to its power set $P(S)$ is just any old collection of ordered pairs $(a,T)$, where $a$ is a member of $S$, and $T$ is a subset of $S$, such that $a$ is only sent to one $T$, and every $a$ in $S$ is sent to some $T$.

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$$ f : A \to 2^A $$ is a mapping $f$, where each element $a$ of a set $A$ maps to an element $B$ of the power set of $A$, which by definition is the set of all subsets of $A$, thus $a$ is mapped to a subset $B$ of the set $A$: $$ f(a) = B, a\in A, \left( B \in 2^A \iff B \subseteq A \right) $$ Why this construction is useful for the proof you study, I can not tell without looking at the proof. (If this is, what you mean with meaning)

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A function is a mapping of one set to another set. That's all. If $A$ and $B$ are sets, and $f:A\to B$ is a function, then this means that $f$ maps every element of $A$ to some element of $B$.

In the case when $B$ is equal to the power set of $A$, this means that $f$ is simply a function that maps each element of $A$ to some element of $P(A)$. In this particular case, this means that $f$ maps each element of $A$ to some subset of $A$.

There is no meaning other than this, functions are simple objects. We then ADD a meaning to the functions when we define that two sets are equipotent if and only if there exists a bijection between them. Still the functions themselves are simple mappings, telling, for each $a\in A$, what $f(a)$ is.

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As previous answers indicate, there is no necessary "meaning", a function is simply a mapping of one set into another, and the second one can be the power set of the first just as it can be another set. Nevertheless, here are two examples which can make it a bit less confusing for you. MATLAB has a function named primes, in which primes(n) returns a list of all prime numbers up to n. So this is a function from the set of natural number to its power set, $primes(5)={2,3,5}$. Another example: take $A$ to be the set of professional football players, and define $f$ to be the function that for each football player returns all players in his team, again this is a function from the set $A$ to its power set.

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