Definition talk:Image (Set Theory)/Relation/Element/Singleton

Can we attempt to incorporate some of this material into existing pages on the appropriate subjects?

This page, for example, holds the same information as Definition:Image of Subset under Relation but in a different language. I understand that Takeuti (and his colleague - please remember to cite this work accurately - it might be worth considering amending your template) takes a different approach from the conventional one (e.g. failing to specify what sort of objects are being referenced when writing a definition; $x$ and $y$ are presumably sets or classes or something, and using letters like $F$ for a general relation when the usual technique on this site so far has been to use $\mathcal R$), but it's probably worth seeing whether some sort of consolidation is possible.

Also see Definition:Value for the appropriate disambiguation page, in which it is apparent that this concept has already been homed. --prime mover 11:31, 10 August 2012 (UTC)


 * Sorry - I see you've already updated your Takeuti template. Good stuff ... --prime mover 11:46, 10 August 2012 (UTC)


 * Just a note. The current definition Definition:Image of Element under Mapping defines $f(s) = \{ t : ( s,t ) \in f \}$, so in this case, that would be $f(s) = \{ t \}$ if $t$ is the unique element such that $( s,t ) \in f$.  This definition makes it $f(s) = t$, which is what seems to be intended by the definition. --Andrew Salmon 17:26, 10 August 2012 (UTC)


 * But is $F$ a mapping or a relation? I think I can understand what this page is trying to say but it doesn't say it very clearly. And, as I say, its intent is already covered in Definition:Image of Element under Mapping - or is it? What are $x$ and $y$? --prime mover 17:32, 10 August 2012 (UTC)


 * Intent is already covered there, but the definition is wrong there. As for $x$ and $y$, $x$ and $y$ are the same as $s$ and $t$.  This page would more aptly be titled Definition:Value of Relation or something, since $F$ is a relation. --Andrew Salmon 17:35, 10 August 2012 (UTC)


 * I have tentatively fixed the definition there. --Andrew Salmon 17:45, 10 August 2012 (UTC)

There's a danger in obfuscating the intent of the definition with a wealth of technical details. While I see the point in making a rigorous definition in order to justify the existence of an object based on teh ZF axioms or whatever, there exists the danger of losing the simple workaday definitions of mappings and relations that are understood by people working at a more basic level. That was what I was trying to come down to in the definition in the original page, and I'm not sure that has been captured.

In fact I'm not sure it's even correct, unless I'm missing something. The original page said:

"If $\mathcal R \left ({s}\right)$ for some $s \in S$ (or $\mathcal R \left ({S_1}\right)$ for some $S_1 \subseteq S$) has only one element $t \in T$, then we can write: ..."

whereas the new page states that:

"$\mathcal R \left({s}\right)$ is the unique $t$ such that $s \mathcal R t$."

... which it is not at all. $\mathcal R \left({s}\right)$ is a relation and always equals $\left\{{t \in T: \left({s, t}\right) \in \mathcal R}\right\}$.

It's only when $\left|{\mathcal R \left({s}\right)}\right| = 1$ that $\mathcal R \left({s}\right) = \left\{{t}\right\}$, and in those cases it is often the case that (by abuse of notation, probably) we can write $\mathcal R \left({s}\right) = t$.

In the case of mappings, then $\left|{f \left({s}\right)}\right| = 1$ by definition of $f$ being a mapping. In that case you just don't see $f \left({s}\right) = \left\{{t}\right\}$.

But what you do see is $f \left({A}\right) = B$ where $A$ and $B$ are subsets of $S$ and $T$. This is seen in topology all the time.

I don't know how "correct" it is, but this is how most of the elementary texts I've seen treat this subject. --prime mover 19:23, 10 August 2012 (UTC)


 * I freely admit that the new definition that I have given is useful only for establishing $\mathcal R \left({s}\right)$ as a definitional abbreviation equal to $t$ in exactly those cases when we may write $\mathcal R \left({s}\right) = t$ (a slight abuse of notation to use image and function value the same way; many books don't even use $f(x)$ but instead use $( f " x )$ for the image of $x$ under $f$ and $( f ` x )$ for the function value). The motivation for the new definitions are more formalistic and may even obfuscate the existing definitions.  They are not as much to enhance the understanding of what "$f$ of $x$" means to the reader but rather how that behavior might be established by a definitional abbreviation.  Part of my motivation is also for completeness: I want to be able to finish $\S 6$, which requires the introduction of this definition.  So you can do what you want with this page... --Andrew Salmon 20:11, 10 August 2012 (UTC)


 * Sorry to interfere. The first two lines of the first definition (as in the very first thing the reader will read) state:


 * "Let $\mathcal R$ be a relation.


 * $\mathcal R \left({s}\right)$ is the unique $t$ such that $s \mathcal R t$."


 * Consider the empty relation on $\{1, 2\}$ and tell me please what $\mathcal{R}(1)$ is.--Jshflynn 20:20, 10 August 2012 (UTC)


 * If there is no "unique" $t$ such that $s \mathcal R t$, most definitions, (such as the second one given) will simply default to $\varnothing$. So $\mathcal R (1) = \varnothing$. --Andrew Salmon 20:26, 10 August 2012 (UTC)

Notation
Is that the notation that is really used? Or is $R'$ and $R''$ what is actually intended? Because if it is actually rendered as such (and not just inaccurately rendered), then I suggest for the purposes of consistency on this website, it is left as an example of an alternative notation on this page, and not actually used anywhere, because (sorry but) it looks bizarre (to me) and horribly non-intuitive. --prime mover 10:33, 11 August 2012 (UTC)


 * $R'$ is sort of what is intended, but the $'$ looks like an actual quote (looks more curly), not like a derivative. I have seen this notation in Principia Mathematica, Takeuti/Zaring, Quine's book, etc. --Andrew Salmon 19:24, 11 August 2012 (UTC)


 * What's everyone else think? IMO this notation is suboptimal. At the very least we should attempt to find an appropriate tag to use to render the symbol properly. And we also need to make very sure we link to the appropriate page with exactitude when we use these symbols, because (despite their provenance) they seem pretty obscure.


 * I'm interested in finding out why it's important to need the value of an element under a relation in the first place; being as there is no guarantee that the value is going to be a single element unless the relation is specified as many-to-one, it seems a bit clumsy to me. --prime mover 20:30, 11 August 2012 (UTC)


 * Nah. It's not that important really.  It's just getting something for free.  If all mappings are relations, why not just define it for any relation? --Andrew Salmon 21:30, 11 August 2012 (UTC)