1

I am currently a M.S. of applied mathematics student and a working TA for an axiomatic set theory course. I have helped teach this course in the past, however in my graduate program, the TA usually teaches a lot of the course to gain experience. We are currently covering maps and I had some questions about common terminology and phrases when talking about functions.

To begin I usually start with the following definition:

Let $A$ and $B$ be sets. A map $\phi : A \to B$ is a relation such that for each $a \in A$ there exists a unique $b \in B$ such that $(a,b) \in \phi$.

Then I follow with this remark:

Remark: From the definition of a map if $\phi : A \to B$, the $\phi$ can be viewed as an assignment, to each element $a \in A$, of a unique element $b \in B$.

This sits well, usually, since before we start even talking about sets we cover propositional and predicate logic and the students are familiar with the concept of assigning a truth value to a predicate of some variable(s).

Here is where I start to have questions: in many mathematics textbooks for set theory, sometimes one reads "the map $\phi$ sends/maps/associates $a$ to $b$". While in others, such as the ones I have based my teaching style after, one reads "a map $\phi$ assigns to every $a \in A$ exactly one $b \in B$." Are all these sentences equivalent? Meaning if a student asks if saying "the map $\phi$ sends/maps/associates $a$ to $b$ and "a map $\phi$ assigns to every $a \in A$ exactly one $b \in B$" are interchangeable I can answer with: "yes, they are all just different ways of talking about maps". I understand the definition of a map is unambiguous and the important take away is that $\phi$ is a relation such that $(a,b) \in \phi$ means "$a$ is related to $b$" - but I wanted to clarify some common vernacular.

Disclaimer: This was originally posted on Mathematics Educators.SE but I felt I would get more precise answers from the community here.

Taylor Rendon
  • 2,276
  • 1
    The definition of what a function is requires the quantifiers. A specific example on the other hand, is explicitly given, usually by a formula of the assignment, without any quantifiers (but setting the domain and codomain). – Berci Aug 26 '20 at 23:59
  • @Berci - I agree. However, do you agree that saying "the map $\phi$ sends/maps/associates $a$ to $b$" can be used interchangeably with saying "the map $\phi$ assigns the element $b$ to $a$"? As in, they are equivalent terminology/jargon when talking about maps. – Taylor Rendon Aug 27 '20 at 00:08
  • 1
    To this question the answer is yes, these are simply synonymous English terminologies. – Berci Aug 27 '20 at 00:13
  • @Berci - I don't believe assign is a synonym to associate, perhaps you meant they are synonymous when speaking about maps? – Taylor Rendon Aug 27 '20 at 00:20
  • 1
    The phrase "the map sends $a$ to $b$" doesn't stand on its own, without additional context, because what are $a$ and $b$? The phrase "the map $\phi$ assigns to every $a \in A$ exactly one $b \in B$" is perfectly clear, though. – littleO Aug 27 '20 at 00:34
  • @littleO - See, so if we added a little more context and said "the map $\phi$ sends/maps/associates every $a \in A$ to exactly one $b \in B$, would this be synonymous (in the context of maps of course) to saying "the map $\phi$ assigns to every $a \in A$ exactly one $b \in B$"? – Taylor Rendon Aug 27 '20 at 00:39
  • 1
    Yeah, that would be synonymous. – littleO Aug 27 '20 at 00:43
  • @littleO - Revisiting this - I never asked exactly why these sentences would be synonymous? Is it just because mathematicians agree that they are synonymous? – Taylor Rendon Aug 29 '20 at 02:48
  • 1
    I would say that words like "assigns" and "associates" are not "official" math terms and they are not usually given precise definitions. If you press a mathematician to say these things more precisely, then they will simply stop using those words, and just give the formal definition of a function. So when they use those words they are being a little informal. And from my experience of what mathematicians mean when they use these terms informally, I can tell that those phrases would be synonymous. (They had better be synonymous, because they're all supposed to agree with the formal definition.) – littleO Aug 29 '20 at 04:10

1 Answers1

1

I think the confusion comes from not always clearly distinguishing between a function $\phi:A\to B$ and the graph of $\phi$, which is the set $\{(a,\phi (a)): a\in A\}$. Specifically, a function specifies its codomain while its graph does not.

For example, let $g:\mathbb{R}\to \mathbb{R}, g(x)=|x|+1$ and $h:\mathbb{R}\to \mathbb{R}^+, h(x)=|x|+1$. Although $g(x)=h(x)$, the functions $g$ and $h$ are different since they have different codomains.

Saying that "the map sends $a$ to $b$" seems to refer to the map's graph rather than the map itself, although the distinction is not usually important.

One way to define a function is as an ordered triple $(X,f,Y)$ where $f \subseteq X\times Y$ satisfying for all $x\in X$ there exists a unique $y\in Y$ such that $(x,y)\in f$.

See Difference between a function and a graph of a function?

Mithrandir
  • 948