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begin quote ita"
CHAPTER I [:] Notions from Set Theory [...] 3. FUNCTIONS. "end quote
These pages assume that the reader has the book and is reading along.    
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Please read these notes from the beginning to here and then the first 3 sentences of paragraph 7 of 3.

FUNCTIONS THAT ARE:         ONE-ONE,         ONTO,         AND         ONE-ONE ONTO

begin quote ita-1-3-7-pg10 "
A function   f : XY   is called [...] one-one [...] if different elements of   X   correspond under   f   to different elements of   Y,   that is if     only if   .   A function   f : XY   is called   onto   if each element of   Y   corresponds under   f   to some elemnt of   X,   that is if each    is of the form     for some   .   If   f : XY   is both one-one and onto it is called   one-one onto   [...].
"end quote

At the beginning of the third paragraph of his preface, Rosenlicht says that he has eliminated many synonyms, and the [...] above indicate that I have eliminated a couple more.   As was the case with the composition of functions, I find that these very simple ideas can be surprisingly confusing. Therefore, I think this is a very good place to make a detailed connection between my learning of German and the learning of mathematics by people who feel that they can't do such learning.

My German grammar book shows the following rule: "All neuter nouns and almost all masculine nouns require only one ending in the singular. This is -s or -es, which is added to the noun to form the genitive. (It is true that some masculine and neuter nouns that are only one syllable long can add -e to form the dative singular, but this ending is optional and can be ignored.)" [Stern & Bleiler, page 17]   This and four other noun rules, as well as five descriptions of noun groups, seemed simple enough, but I had no feeling of understanding at the first twenty readings or so. It all seemed to go in one eye and out the other until I started using my dictionary to study examples. Then things fell into place, and I could see why mathematics educators say "Go from the concrete to the abstract." However, once I started working examples, I had gone successfully from the abstract to the concrete. Before presenting simple examples for the mathematical terms one-one and onto, I would like to descrbie in detail my experience with the German grammar rule just quoted.

I was listening to, and learning the words of, the German national anthem. It uses the noun Glanze meaning glow. This confused me greatly because it did not seem to be covered by the rules. I thought there must be other rules or irregular declensions of which I had no knowledge. This perception did not change even after I asked a German person about this word and verified that it was the dative case. Eventually, I read again the rule quoted above and the light went on "Ahah! Eureka!" That is that optional -e, which I am sure is not optional in this case since it is in the national anthem. Be that as it may, once I had made this connection between the abstract and the concrete, I had a strong feeling of understanding. In this way also, I think, we can go from the abstract to the concrete in mathematics.

Back to the Mathematics:   I understand the terms   one-one   and   onto   by using the common sense of the names together with thoughts about the sizes of the two sets. First, I will not consider infinite sets here because they are treated in the next section. Therefore, consider the three cases shown below, and imagine the two sets arranged in two columns with line segments from the elements of the first set to the elements of the second set. Imagine that these line segments define the function values.

(1) The first set has more elements than the second one.

It is part of the definition of a function that every element of the first set is associated with one (and only one) of the elements of the second set. Therefore, every element of the first set is at the end of one (and only one) line segment. However, each element of the second set can be at the end of zero, one, or more than one line segments. If all of the elements of the second set are at the end of one or more line segments, then the function is   onto.   If there is at least one element of the second set that is not at the end of any of the line segments, then the function is not onto.   In this case, the function can't be   one-one   which we can see as follows: On the left, after we have associated   A   with   D   and   B   with   E,   the definition of a function forces us to associate   C   with one of the elements of the second set, and both have been used already. But ...

(2) The two sets have the same number of elements.

... on the left here, the function can be   one-one onto.   However, on the right we see that the function must be either   one-one AND onto   or   not one-one AND not onto.

(3) The first set has less elements than the second.

Here, the function must be   not onto   because, again by the definition of a function, each element of the first set must be at the end of exactly one line segment. We see on the left that the function can be   one-one   but (on the right) the function can also be   not one-one.

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