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Below is a solution to the question on how many functions exist from $S$ into $T$. However, I do not understand the solution; I do not understand the sentence "Any function from S into T must be a function where codomain must not be equal to range." Why though? Why can't I have both elements of T in the range? It seems like a false statement to me...?

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According to the following site http://mymathangels.com/tag/into-function/ a function $f: A \to B$ is 'into' if not all $b \in B$ are images (so there exist $b \in B$ such that for all $a \in A$ we have that $f(a) \neq b$. Therefore, you only have two possibilities: either all elements of $S$ are mapped to 4 or all elements of $S$ are mapped to $5$. (Note that if both $4$ and $5$ would be images of the same map, then we would have that the function is 'onto' and by definition not 'into').

The other question (the number of maps from $T$ to $S$) is, in my opinion, much more interesting, since it does not need this special (non-standard) notion of 'into'. Can you find the number of maps from $T$ to $S$?

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    Thank you - that answered the question perfectly. ALthough, this guy here refers to an "into" function as if it is synonymous with one-to-one...and I somewhat recall learning it that way, once years ago. http://mathforum.org/library/drmath/view/52454.html. Interesting. But I think without these special notions of "into," the answer should be 8. – 2017-02-26
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    I had to look it up myself and found all kind of definitions! I think the point is that either they should make it a clear definition in your lecture notes or in the problem statement. With the regular notion of 'function' it is (indeed) 8. From $T$ to $S$ you should find (using the same reasoning) 9 different functions. – 2017-02-26
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It seems that by "function from $S$ into $T$" it means a function $f \colon S \rightarrow T$ such that the image $f(S)$ is not the whole of $T$. This is not standard terminology but under this interpretation, the solution makes sense.

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    But why does it make sense? Why is it saying that the image cannot be the whole of T? Can't I maybe find a function that maps elements 1,2 and 3 to the numbers 4 and 5? – 2017-02-26
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    The terminology doesn't make sense... You can definitely find functions from $S$ to $T$ whose image is the whole of $T$. The number of functions from $S$ to $T$ is $2^3 = 8$. I'm just trying to interpret the terminology so that the solution makes sense. A more standard terminology would be "Find the number of functions from $S$ to $T$ which are not onto". – 2017-02-26
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    Oh, I see what you meant now levap. I was going to agree with you but I think "Student's" solution below will answer that the word "into" actually means something specific here. Thanks! – 2017-02-26
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    @PBJ: No problem. I just wanted to warn that this terminology is completely non-standard and is not used in all textbooks or articles (in fact, I have never seen it before) so your objection is quite valid. – 2017-02-26