2

In the book I am reading (Abstract Algebra, Dummit & Foote), the author uses 2 ways to define functions:

$$f(x) = \Box$$ $$x \mapsto \Box$$

It's not that I don't know what they mean - it's that they use both, which leaves me feeling like I am missing something, when a particular choice is used.

For example, just a few lines apart, they write a group action as

$\sigma_{g}: A \rightarrow A$ defined by $\sigma_{g}: a\mapsto g \cdot a$

and a group homomorphism as

$\varphi:G \rightarrow S_{n}$ defined by $\varphi (g) = \sigma_g$

Is there a reason one form would be used over the other?

Ѕᴀᴀᴅ
  • 34,263
Abel
  • 45
  • 3
  • 4
    It's a guess: perhaps Dummit insisted on one way and Foote the other way? – Berci Sep 29 '18 at 16:13
  • @Berci Ah, I wish I had the rep to upvote that :) – Abel Sep 29 '18 at 16:14
  • 2
    I use $x\mapsto \Box$ when I'm talking about a function and don't feel the need to give it a name. That's clearly not what's done here, though. – Arthur Sep 29 '18 at 16:15
  • 3
    There is no underlying difference, but it does represent a slight difference in how you think about the function. In the first case, you are thinking of $g$ acting on the elements of $A$, and then this defining a function; the notation reminds you that you are thinking about what is hapening to each element. In the second you are thinking about the function as doing something to the entire set/group. Moreover, in this particular instance, you have a function that sends elements to functions; so the distinction also helps keep things straight; the functions that are images use the latter. – Arturo Magidin Sep 29 '18 at 16:15
  • 1
    While your original function uses the former. Otherwise, you would write something like $\varphi(g)(a) = g\cdot a$; this way, you write $\varphi(g)\colon a\mapsto g\cdot a$. – Arturo Magidin Sep 29 '18 at 16:17
  • See page 1 bottom. – Mauro ALLEGRANZA Sep 29 '18 at 16:19
  • @MauroALLEGRANZA I'm afraid I don't understand the subtly here :( – Abel Sep 29 '18 at 16:31
  • @ArturoMagidin A little confusing - but I suppose the basic idea of keeping the 2 distinguished might be what they were after. Seems to have backfired a bit (for me at least!) – Abel Sep 29 '18 at 16:34
  • Sometimes you write $y'$, sometimes you write $\frac{dy}{dx}$. – Randall Sep 29 '18 at 16:55
  • FYI, a complete, rigorous notation for a function $f$ is $$f:X\to Y,\quad x\mapsto \ldots$$ where $\ldots$ denotes the value of $f(x)$ in $Y$ for each $x$ in $X$. – Did Sep 29 '18 at 17:56

1 Answers1

0

Long comment ( hope it helps...)

See : Saunders Mac Lane & Garrett Birkhoff, Algebra, AMS (3rd ed., 1991), page 4 :

A function $f$ on a set $S$ to a set $T$ assigns to each element $s$ of $S$ an element $f(s) \in T$, as indicated by the notation

$s \mapsto f(s), \ \ \ \ s \in S$.

The element $f(s)$ may also be written as $fs$ or $f_s$, without parentheses; it is the value of $f$ at the argument $s$. The set $S$ is called the domain of $f$, while $T$ is the codomain. The arrow notation

$f : S \to T \ \ \text {or } \ \ S \stackrel{f}\longrightarrow T$

indicates that $f$ is a function with domain $S$ and codomain $T$. [...] We systematically use the barred arrow to go from argument to value of a function and the straight arrow $S \to T$ to go from domain to codomain.

Mauro ALLEGRANZA
  • 94,169
  • Hey Mauro. The issue isn't so much one of domain / codomain - this is common to both notations I listed. – Abel Sep 30 '18 at 19:48