# What is a one to one relationship that never changes? [closed]

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?

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

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• 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. – angryavian Jul 27 '17 at 20:36
• 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. – user228113 Jul 27 '17 at 20:38
• @angryavian Erm, am not sure. Don't get bogged down in technicalities here, there disposable means to serve the principle. – Jim Jam Jul 27 '17 at 20:38
• @G.Sassatelli isn't this related to functions and/or set theory? – Jim Jam Jul 27 '17 at 20:39
• 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". – user228113 Jul 27 '17 at 20:40

## 3 Answers

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.

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.

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.