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Say you have sets $A$ and $B$, whereby $A$ contains elements $a_1$, $a_2$, and $a_3$, and $B$ contains $b_1$, $b_2$, and $b_3$.

The two sets have a one-to-one relationship in such a way that $a_1$ maps to $b_1$, $a_2$ to $b_2$, and $a_3$ to $b_3$.

What do you call, however, a one-to-one relationship that never changes, i.e, one that's static? So if $a_1$ now maps to $b_2$ and $a_2$ to $b_1$ (with $a_1$ and $b_1$ and $a_2$ and $b_2$ no longer being connected, respectively), it's still a one-to-one relationship, but it's a dynamic one. If this isn't something that's possible (for whatever reason), what type of one-to-one relationship do you call this?

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closed as unclear what you're asking by JMoravitz, Trevor Gunn, José Carlos Santos, JonMark Perry, steven gregory Jul 28 '17 at 7:14

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    $\begingroup$ Each set has different objects, so "never changes" does not really have meaning. I infer that you are probably talking about the subscripts, but in this case you can just talk about maps from $\{1,2,3\}$ to $\{1,2,3\}$, and then a "never changing" relationship would be called the "identity" map. $\endgroup$ – angryavian Jul 27 '17 at 20:36
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    $\begingroup$ You are asking for a concept which appears not to be mathematical. Or, rather, you are not making clear the mathematical meaning of what you are asking. $\endgroup$ – user228113 Jul 27 '17 at 20:38
  • $\begingroup$ @angryavian Erm, am not sure. Don't get bogged down in technicalities here, there disposable means to serve the principle. $\endgroup$ – Jim Jam Jul 27 '17 at 20:38
  • $\begingroup$ @G.Sassatelli isn't this related to functions and/or set theory? $\endgroup$ – Jim Jam Jul 27 '17 at 20:39
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    $\begingroup$ And the point is that, in these terms, there is no such set theoretical concept as "being static". A function is a particular set of pairs, period. Another function is another set of pairs. Nothing "changing". $\endgroup$ – user228113 Jul 27 '17 at 20:40
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There are six different one-to-one correspondences between two sets of three elements $\{a_1,a_2,a_3\}$ and $\{b_1,b_2,b_3\}.$ Here they are: $$ \begin{array}{lll} \left\{\begin{array}{c} a_ 1 \leftrightarrow b_1 \\ a_2 \leftrightarrow b_2 \\ a_2 \leftrightarrow b_3 \end{array} \right\} & \left\{ \begin{array}{c} a_1 \leftrightarrow b_2 \\ a_2 \leftrightarrow b_1 \\ a_3 \leftrightarrow b_3 \end{array} \right\} & \left\{ \begin{array}{c} a_1 \leftrightarrow b_1 \\ a_2 \leftrightarrow b_3 \\ a_3 \leftrightarrow b_2 \end{array} \right\} \\[10pt] \left\{ \begin{array}{c} a_1 \leftrightarrow b_3 \\ a_2 \leftrightarrow b_2 \\ a_3 \leftrightarrow b_1 \end{array} \right\} & \left\{ \begin{array}{c} a_1 \leftrightarrow b_2 \\ a_2 \leftrightarrow b_3 \\ a_3 \leftrightarrow b_1 \end{array} \right\} & \left\{ \begin{array}{c} a_1 \leftrightarrow b_3 \\ a_2 \leftrightarrow b_1 \\ a_3 \leftrightarrow b_2 \end{array} \right\} \end{array} $$

Not one of them ever changes. However, you could consider one of them on Monday, another on Tuesday, another on Wednesday, and so on. You could have a sequence of one-to-one correspondences between these two sets of three elements. Or in some other way you could have a function whose values are within this set of six one-to-one correspondences. But precisely what you have in mind here is unclear beyond that.

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It sounds to me like you are really considering a family $\{\varphi_t\}$ of maps $\varphi_t:A\to B$, and you are requiring $\varphi_t(x)=\varphi_s(x)$ for all times $s,t$ and all $x\in A$. I say that because you speak of what certain elements map to "now" as opposed to earlier. I would just say that these maps are pointwise constant over time, or something to that effect. I'm not aware of any special name for that.

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Mathematicians generally assume that everything is constant over time unless otherwise stated. So the term to use for "a one-to-one relationship that never changes" is just "a one-to-one relationship".

If you want to emphasize that the relationship is constant (which is only useful if you're talking about other things that are "changing"), call it "a constant one-to-one relationship".

By contrast, if you wanted to talk about a one-to-one relationship that does change, you would need to explicitly say that it may change, and specify the variables that the relationship depends on.

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