What does let $F$ denote the set of all functions from $\{1, 2, 3\}$ to $\{1, 2, 3\}$ mean?

Question

Let $F$ denote the set of all functions from $\{1, 2, 3\}$ to $\{1, 2, 3\}$.

I'm supposed to prove that this statement is true or false, $$∀f ∈ F, \;∃g ∈ F\tag i$$ so that $g(f(1)) = 2$ But I'm not sure what $F$ is exactly.

Answer 1

A function from $\{1,2,3\}$ to $\{1,2,3\}$ is a rule or some other procedure that takes any of $1,2,$ or $3$ and gives you back one of $1,2,$ or $3$.

One example is the rule that says

• Always give back the number you got.

Another is:

• Always give back the number 3.

A third is:

• If you got $1$ or $2$, give back $2$, but if you got $3$ give back $1$.

As it happens there are 27 possible behaviors such a function can have, and in this problem the letter $F$ is being used to name the set that contains all 27. This means nothing more nor less than saying that "$f\in F $" is now an abbreviation for "$f$ is one of those 27 functions".

If $f$ is one of these 27 functions, then we write $f(1)$ to mean “the number that $f$ gives back if it gets $1$”.

The claim you are supposed to evaluate is: If $f$ is any one of these 27 functions, is there another one of these 27 functions (or maybe the same one), which we might call $g$, so that $$g(f(1)) = 2?$$

That is, must there always be some function $g$ so that if you give $1$ to $f$, then take what $f$ gives back and give it to $g$, then $ g$ always gives you back $2$?

If you think so, you should describe a way, for each $f$, of constructing the corresponding function $g$ that makes $g(f(1)) = 2$. If not, you should come up with an example function $f$ for which there is no $g$ that makes $g(f(1)) = 2$, and an argument why there isn't.

Answer 2

If $F$ is the set of all functions from $\{1,2,3\}$ to $\{1,2,3\}$ then $F=\{f | f:\{1,2,3\}\rightarrow \{1,2,3\}\}$.

Our definition of a function is that the function $f$ maps every element from the first set to one element of the second set. So, for example, $h\in F$ where $h:\{1,2,3\}\rightarrow \{1,2,3\}$ and $h$ is defined as $h(1)= 2$, $h(2)= 3$, and $h(3)= 1$.

So, for all functions $f(1)$ may have the value of 1, 2 or 3. Does there exist a function $g\in F$ such that $g(1)=2$, $g(2)= 2$ or $g(3)=2$?