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Does this property have a name?

Sid
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  • I think that is a property of all functions – miracle173 Sep 26 '13 at 21:01
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    It's the definition of function that there is only one $f(x_1)$. Did you mean $f(x_1) = f(x_2) \Rightarrow x_1 = x_2$? – Daniel Fischer Sep 26 '13 at 21:01
  • I am aware that this is the property of all functions, but I would like to know if this property has a name. – Sid Sep 26 '13 at 21:02
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    In complex analysis, we may sometimes say it is "single-valued". – GEdgar Sep 26 '13 at 21:10
  • @GEdgar: I have seen also univalued, as the opposite of multivalued function. There may be a risk that a reader could interprets this as meaning constant. – Henry Sep 26 '13 at 21:31
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    Typically the first year calculus book refer to this property as "every function satisfies the Vertical Line test"... – N. S. Sep 26 '13 at 22:14

4 Answers4

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This is just inherent in the definition of a function, which cannot take one input to two distinct outputs.

As for a name for it, I think it's often referred to as a function being "well-defined."

Devlin Mallory
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  • I know, but what is this property called? – Sid Sep 26 '13 at 21:02
  • "Well defined" is equivalent to "fully" and "uniquely" defined, as long as this page is correct at this: http://en.wikipedia.org/wiki/Finitary_relation#Analogy_with_functions – ftfish Sep 26 '13 at 21:53
  • I am sorry, but the Wikipedia article you cite is not using standard English-Math(s) terminology. – Rob Arthan Sep 26 '13 at 22:28
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(The relation is) "uniquely defined" or "tubular at $X$" where $X$ is the left side of the relation. See this wiki page. The former seems preferred by many, including me.

As @RobArthan indicates, there's yet another name: (The relation is) "functional". See this site. The other property for a relation to be a function (fully defined) is called "entire".

FYI:

A function is a uniquely and fully defined relation.

A relation $R$ is well defined $\Leftrightarrow$ $R$ is a function $\Leftrightarrow$ $R$ is both uniquely and fully defined.

"Uniquely defined" is called "rechtseindeutig" in German, which is literally "right unique".

"Fully defined" is called "linkstotal" in German, which is "left total".

ftfish
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  • The wiki page does not say that a relation $R\subset X \times Y$ is uniquely defined or tubular but it is uniquely defined on X or tubular on X or on Y. – miracle173 Sep 26 '13 at 21:35
  • @miracle173 Thank you. I didn't notice that. Edited. – ftfish Sep 26 '13 at 21:41
  • I am impressed that 3 points and a tick go to an answer that explicitly says that it doesn't give an answer. – Rob Arthan Sep 26 '13 at 22:04
  • @RobArthan I think I gave the answer in Edit. Hope you found it. Well, I can make it clearer to the tl;dr people. Thanks anyways. – ftfish Sep 26 '13 at 22:05
  • @RobArthan Done. Hope you like it the way it is now. – ftfish Sep 26 '13 at 22:09
  • Sorry, but I don"t think "fully defined" adds anything. (I am taking the question as being about idiomatic use of English in mathematics and the idioms you are suggesting don't work for me). – Rob Arthan Sep 26 '13 at 22:16
  • I think it was a good answer. But it wasn't necessary to remove "richtseindeutig" and "linkstotal" from your answer, I think it was very interesting to hear the German perspective. – Sid Sep 26 '13 at 22:16
  • @Sid added back, thanks! – ftfish Sep 26 '13 at 22:21
  • @RobArthan but the object in question is a relation, not already a function. See the "FYI" part please. – ftfish Sep 26 '13 at 22:22
  • @sid & ftfish: I did not mean to appear denigratory about ftfish's original answer. I really was impressed that an answer that explicitly says it isn't an answer can be a useful and interesting answer :-) – Rob Arthan Sep 26 '13 at 22:25
  • @RobArthan I'm sorry for my very first answer to this question, which in fact said I didn't know the answer. But it is the edited answer which did include an answer (or at least I think is) that got accepted. – ftfish Sep 26 '13 at 22:28
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In the standard set-theoretic way of defining functions, a function is a binary relation (i.e., a subset of $X \times Y$ for some sets $X$ and $Y$) with the property that $(x,y),(x,y') \in R \implies y = y'$, for all $(x,y)$ and $(x,y')$. So the property you mention is just called "is a function".

Magdiragdag
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A typical English-Maths speaking (or English-Math speaking) audience will immediately understand "R is a functional relation" or "R is a single-valued relation" or (as Peter suggests) just "R is a function". As a native English-Maths speaker, I would not understand "R is tubular", and would only understand "R is right-unique" because I know the German "richtseindeutig".

Rob Arthan
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  • It's good that someone notes that "tubular" is rarely used. But I think this definition is meaningful since a "functional relation" does not exactly mean the same thing that I was asking for, since it also implies that the relation is "fully defined" which "tubular at X" does not. – Sid Sep 26 '13 at 22:22
  • "Functional relation" does not imply "fully defined" (and "fully defined" is not standard English-Math(s) terminology). – Rob Arthan Sep 26 '13 at 22:32
  • @RobArthan By "functional relation" you mean partial function? I'm surprised if so. – ftfish Sep 26 '13 at 22:35
  • @ftfish: In standard English-Math(s) usage you just say "$R$ is a functional relation". That statement doesn't mention a possible domain of definition for $R$, so the information that distinguishes partial and total functions is not available in the statement. See http://www.encyclopediaofmath.org/index.php/Functional_relation – Rob Arthan Sep 26 '13 at 22:40
  • @RobArthan What is then a functional relation between $X$ and $Y$? A partial function? From the name I would assume a function... – ftfish Sep 26 '13 at 22:41
  • @ftfish: What's in a name? If I read "functional relation between $X$ and $Y$", I would not be surprised if the author intended to include functions that are not total on $X$. – Rob Arthan Sep 26 '13 at 22:55
  • @RobArthan The link you gave does say such a thing is a partial function. This site says "functional" and "entire" are other names for those two properties: http://ncatlab.org/nlab/show/functional+relation . So you're probably right that it doesn't imply $R$ is fully defined, which really surprises me. I added this system of terminology to my answer, hope you don't mind. – ftfish Sep 26 '13 at 23:03