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I am trying to struggle through some basic math textbooks and I noticed that in definitions they usually use the following phrasing: "A function $f:X \to Y$ is called one-to-one if..." I was wondering if there was a deep reason why they don't say "A function $f:X \to Y$ is one-to-one if..." or if it could be said either way.

Mauro ALLEGRANZA
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Adam Shai
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    No; we simply "name" it; we "baptize" that specified kind of function with the name one-to-one. – Mauro ALLEGRANZA Mar 18 '18 at 16:39
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    Without the "is called" and without further context your phrasing could be stating a theorem about one-to-one functions rather than a definition. I agree that Inside a formal definition you could safely omit "is called", but it's better to be safe than rely on context. – Ethan Bolker Mar 18 '18 at 16:43
  • In my experience I have found most textbooks do so "is" rather than "is called". But shouldn't it be obvious that if we are defining an expression for the first time that the phrase will have no meaning until it is defined. To say A function is one-to-one when one-to-one is meaningless is meaningless. And if one-to-one is meaningful then to say a function "is" one-to-one when it fulfills the meaning is redundant. A definition is neither of those cases. A definition is a declaration that "this is what a phrase means". Sure the deep difference of that is obvious. – fleablood Mar 18 '18 at 16:44
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    If the statement is floating amid the general text of an article, "is called one-to-one" can help highlight that you're making a definition. You can achieve the same end with a structure like "Definition. A function [...] is one-to-one if ..". Generally, you can do what you want, so long as your intention is clear to the reader. After all, you might also want to use emphasis for descriptive phrases that aren't definitional: "a function that does this-or-that is weird ..." – Blue Mar 18 '18 at 16:44
  • It signals that the statement is a definition, rather than a theorem. – Bram28 Mar 18 '18 at 16:45
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    http://www.cs.utoronto.ca/~chechik/courses/csc324/white.html – Lee Mosher Mar 18 '18 at 16:48
  • @LeeMosher He who got Carroll's message got it all ;-) – Andreas Mar 18 '18 at 17:33

2 Answers2

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It's to distinguish the act of introducing a name from stating a claim.

If I write

A foo is blah iff it gleens,

it's unclear whether I'm defining the notion of a "blah foo" or if I'm claiming that I can prove that the blah foos are exactly the ones which gleen. By contrast, if I write

A foo is called blah iff it gleens,

it's clear that what I mean is that I'm introducing the term "blah" here. Basically, this is letting the reader know that they haven't missed the definition of "$blah$" earlier in the text, and that I'm not tacitly assuming that they already know what "blah" means. Note that in mathematical writing, we often do state "immediate" observations without proof (e.g. "A function is injective iff it has a left inverse"), so confusion is definitely possible here.

(Unfortunately this isn't universally used.)

Noah Schweber
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"is called" is a statement about phrases and meaning. "is" is a statement about actual existence.

To state: "A function, $f:X\to Y$ is one-to-one if for any $x\in X; y\in Y$ so that $f(x)=y$ then there is no other $z \in X; z \ne x$ so that $f(z) = f(x)$" when "one-to-one" has not been defined is meaningless because "one-to-one" is meaningless (at this time).

To state: "A function, $f:X\to Y$ is one-to-one if for any $x\in X; y\in Y$ so that $f(x)=y$ then there is no other $z \in X; z \ne x$ so that $f(z) = f(x)$" when "one-to-one" has been defined is redundant because that is what one-to-one means so this statement is pointless and repetitive.

To define what "one-to-one" means the first time we must make a statement that says: "We will use a phrase $X$ to mean this particular condition $C$". But this is a statement about the phrase "one-to-one" and declaring its meaning. It isn't actually declaring anything about the universe that isn't already known (merely that will will be referring to a concept by a specific term.)

fleablood
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