Older textbook, definitions incoherent with contemporary theory

Question

I have an Italian textbook from 1988 (Nicola Fedele), I encountered a definition of codomain incoherent with the contemporary one. It defines the codomain as:

To denote a function f of set X ⊆ S in T, the following notation is used:
f : X ⊆ S → T
...
The set X is named definition set or also domain of the function f; while the subset of T composed of the elements in relation to the elements of X thru the function f is named values set or also codomain of the function f, and is denoted with the symbol f(X).

Here the codomain is defined as what modern math defines as image.

I wasn't expecting such an inconsistency from a 1988 textbook, as I'm also reading a textbook from 1994 which presents the "correct" definition.
My question is, if inferrable, if it is plausible that I'll encounter several other quirks from this textbook, and I should rather purchase a different one, or if it's likely that this is a rare exception and I can turn a blind eye at it. My objective is to learn contemporary math, I'm not interested in older theories.

Answer 1

You need not be scared by terms used unconventionally as long as those terms are unambiguously defined by the Author. Yes, today codomain is a term that all agree to define in the same ways, but still some 30 years ago in non-English speaking countries codomain could be defined as the image or as the target set depending on author's preference. The same kind of ambiguity that today still persists for range in English speaking countries.

As to biunivocal that you say in a comment, it is not really clear what you mean, because this term is not very much used internationally in math. Probably you mean one-to-one. Note that a one-to-one function is said an injective function, while a function that is one-to-one corrispondence is said a bijective function. So be warned about such a subtlety. In Italian one-to-one is not usually translated word by word, rather it is translated with a word that indeed resemble "biunivocal", but there is no difference between a "biunivocal" function and a function that is a "biunivocal" corrispondence in Italian, they both mean bijective function. So stick to injective and bijective terms, that have no translation problems.

But again this is not important, as long as a definition is given with no ambiguity. If you deeply learn concepts, you will not feel uneasy changing terminology when a new context occurs.

Furthermore, with the notation: $$f\colon X\subseteq S\to T$$ the Author says what he means, that is, he defines it unambiguously (reread the definition, if not clear).

I've never read Nicola Fedele, but I know that he is an authority, pupil of Federico Cafiero. So I would not discard his book (still in widespread use) just for the unconventional terminology.

Just as an example Herstein in his beautiful "Topics in Algebra" second edition 1975 had been using isomorphism into and isomorphism onto instead of the now common monomorphism and isomorphism, respectively. But this is a book that still is worth reading, even though at the beginning one feels uneasy because of that: at least it helps you to develop a critical approach when reading math in terms of paying the utmost attention to every single word.

Answer 2

Analysis of the posted text is complicated by it being a faulty translation from the original Italian. However, some features appear to be a fair representation of the author's writing, and some of those are nonstandard. The notation $f : X ⊆ S → T$ is nonstandard. We would normally write $f : X → T$, with $X ⊆ S$ either being obvious from the context or stated separately. In English, $f(X)$ is called the range (or image, in some contexts) of $f$, and $T$ is the codomain. Calling the range "codomain" is simply contrary to normal usage. The term domain is correctly used, but definition set has little currency in English. The term value set (or the grammatically incorrect form values set) is also unconventional in English.

In mathematical writing, some unconventiality may improve on the standard form, or may be justified by the particular context; but generally it makes life harder for readers—and for students in particular. It's difficult to assess whether this author's terminological creativity will cause real problems of understanding for you. While I can see some logic in it, I wouldn't recommend teaching students usage that conflicts with what they will encounter elsewhere.

In English we might write "a function from a set $X$ to a set $T$, where $X\subseteq S$ " or "a function on a set $X$ with values in a set $T$, where $X\subseteq S$ ". The abbreviated form "a function from a set $X\subseteq S$ to a set $T$ " would perhaps be considered acceptable by enough people, but the purely symbolic form $f : X ⊆ S → T$ is not normal. Viewed syntactically, the arrow points from the $S$ to the $T$. But $S$ has nothing to do with the function $f$; it just happens to be the case that $X\subseteq S$. The notation seems to conflate function with partial function.

I would edit the rest of the translated text to:

"The set $X$ is called the definition set or domain of the function $f$. The subset of $T$ comprising the values $f(x)$ ($x\in X$) is called the value set or codomain [sic] of $f$ and is denoted by $f(X)$."