Looking for a rule about sets and the relation or mapping between them

[SenatusPopulusqueRomanus]

I'm looking for a rule to cite about how a mapping or relation between elements of two sets implies a relation originating from the superset of those sets of elements.
I understand that typically, if elements of two different sets are paired by a relation or mapping, set theorists say that this indicates a subset that contains those relations.
But in cases where any element of one set can be mapped/paired to any element of the other set, it seems to me this proves that the relation resides in the superset of both sets... because the relation itself isn't constrained to any particular pairing. The relation in question obviously contained the total elements of both sets to start with, hence its ability to pair any element to any number of other elements.

Is there a rule about how a set can be proven to be true, if we are given or assume information about the pairings of elements between two sets?
Is there a rule or a recommended resource I can find where the rules are laid out?
Many thanks for any help.

 

[pka]

Is there a rule about how a set can be proven to be true, if we are given or assume information about the pairings of elements between two sets?
Is there a rule or a recommended resource I can find where the rules are laid out?

The answer to that is a sound NO. Sets are not true or false!
Relations and functions are two different different concepts.
Yes they are both sets of ordered pairs: subsets of a Cartesian product of sets.
A relation is a subset of [imath]S\times S[/imath] i.e. a cross product of a set with itself.
A function is a subset of [imath]S\times T[/imath] i.e. a cross product of sets where [imath]S\ne T \text{ or }S=T[/imath].
Functions are more restrictive: the statement that [imath]f:S\mapsto T[/imath] is a function means:

1. [imath]f\subseteq S\times T[/imath]
2. [imath]\text{If } x\in S \text{ then }\exists y \in T\text{ such that }(x,y)\in f. [/imath]
3. No two pairs in [imath]f[/imath] have the same first term. (i.e. the vertical line test).

Now with that background please rephrase your question.

 

[SenatusPopulusqueRomanus]

A relation is a subset of [imath]S\times S[/imath] i.e. a cross product of a set with itself.

Thanks, how is this derived? Or is it simply defined?

 

[blamocur]

It is defined.

 

[Dr.Peterson]

Thanks, how is this derived? Or is it simply defined?

The definition pka gave is for a relation on a single set. There is also a more general concept of relation between two sets, of which a function is an example:

Binary relation - Wikipedia

en.wikipedia.org

That is presumably what you are talking about.

But I can't make any sense of your question itself. Please explain more fully (perhaps with examples and sources) what you are saying about supersets and subsets, and what you mean by a set being true or false.

 

[SenatusPopulusqueRomanus]

It is defined.

Thanks, in this case, I think the original definition was erroneous. In the definition given, the "⊆" is contradictory, because the elements of X are all related to each other by being in the same set to start with. The relation has either the same (e.g. a duality), or greater standing as the set itself.

 

[SenatusPopulusqueRomanus]

The definition pka gave is for a relation on a single set. There is also a more general concept of relation between two sets, of which a function is an example:

In my question I said two sets, but the rule or proof I am looking for should be applicable to both.

Binary relation - Wikipedia

en.wikipedia.org

That is presumably what you are talking about.

Yes, I'm talking about relations in the most generic sense:
The reason I originally said "mapping or relation" is because for a relation to be able to be "many-to-many" is an important aspect here, and I've only found resources that point out this capacity in the case of mappings (but I now see that in the Wikipedia article blamocur posted, this is included among properties of relations), so I wanted to mention mappings explicitly.

Please explain more fully (perhaps with examples and sources)

Sources and examples are what I'm asking for. I said "if elements of two different sets are paired by a relation or mapping, set theorists say that this indicates a subset that contains those relations.". I'm not sure how to explain more fully, other than to say that the definition given here: https://en.wikipedia.org/wiki/Relation_(mathematics)#Definition seems to be as far as I can get to the actual reasoning why set theorists say that a relation is always a subset of the set or sets it relates the elements of.

what you are saying about supersets and subsets

I don't think there is a source within mathematics (that is what I posted this thread to ask), there is only the reasoning I gave. I am saying that if the elements of two sets can be ordered into a relation / paired in any way, then this indicates that the relation isn't a subset. To research the idea more fully, I came here to ask what the official proof is that a relation is always a subset of the sets of the elements it relates. If the relation R always being a subset or some set is just a definitive assertion with no explicit proof, it's possible there are new discoveries to be made.

and what you mean by a set being true or false.

I mean that set is indicated by / produced by / a fact proven from, some operation or circumstance.

 

[blamocur]

In the definition given, the "⊆" is contradictory, because the elements of X are all related to each other by being in the same set to start with.

I don't see any contradiction here -- can you elaborate?
Thanks.

 

[SenatusPopulusqueRomanus]

I don't see any contradiction here -- can you elaborate?

Because the elements of X are all related to each other by being in the same set to start with. There is a relation that contains all of those elements which allows them to be in the same set -- that relation isn't a subset. If there were no relation in which the elements weren't presuppositionally related, they'd bear no relation: not even a common set in which to be situated.

 

[blamocur]

Because the elements of X are all related to each other by being in the same set to start with. There is a relation that contains all of those elements which allows them to be in the same set -- that relation isn't a subset. If there were no relation in which the elements weren't presuppositionally related, they'd bear no relation: not even a common set in which to be situated.

This sounds more philosophical than mathematical explanation. Being in the same set is just one ("full") relation among many possible relations. A general relation [imath]R[/imath] is defined as a subset of [imath]S\times S[/imath], i.e., [imath]R\subset S\times S[/imath], but in the case of the "full" relation we have [imath]R = S\times S[/imath].

For example consider the case of a set [imath]S[/imath] with 5 numbers : [imath]S = \{1,2,3,4,5\}[/imath] the set [imath]S\times S[/imath] (which also represents the "full" relation) has 25 elements (pairs of numbers), but the relation [imath]x < y[/imath] has only 10 pairs.

 

[SenatusPopulusqueRomanus]

but in the case of the "full" relation we have R=S×SR = S\times SR=S×S.

For example consider the case of a set SSS with 5 numbers : S={1,2,3,4,5}S = \{1,2,3,4,5\}S={1,2,3,4,5} the set S×SS\times SS×S (which also represents the "full" relation) has 25 elements (pairs of numbers), but the relation x<yx < yx<y has only 10 pairs.

Thank you, is there a source I can cite for what full relations are?

 

[blamocur]

Thank you, is there a source I can cite for what full relations are?

I used quotes for "full" because I've made up the term myself. I did not see it being used elsewhere.
Note: you can find definitions of complete or total relations, but they are not "full".

 

[Dr.Peterson]

Sources and examples are what I'm asking for. I said "if elements of two different sets are paired by a relation or mapping, set theorists say that this indicates a subset that contains those relations.". I'm not sure how to explain more fully, other than to say that the definition given here: https://en.wikipedia.org/wiki/Relation_(mathematics)#Definition seems to be as far as I can get to the actual reasoning why set theorists say that a relation is always a subset of the set or sets it relates the elements of.

I was looking for a source of your claim that "set theorists say that this indicates a subset that contains those relations". I think you are probably misunderstanding a number of things, so it will be helpful to see what was actually said that you are asking about.

Your wording here makes no sense in terms of the actual definition of a relation; a relation is a subset of the Cartesian product of two sets, by definition -- not a subset of the set(s) being related.

I don't think there is a source within mathematics (that is what I posted this thread to ask), there is only the reasoning I gave.

If no one actually says this, then it would appear to be a figment of your own imagination, and there is nothing to be said about it mathematically.

I am saying that if the elements of two sets can be ordered into a relation / paired in any way, then this indicates that the relation isn't a subset. To research the idea more fully, I came here to ask what the official proof is that a relation is always a subset of the sets of the elements it relates. If the relation R always being a subset or some set is just a definitive assertion with no explicit proof, it's possible there are new discoveries to be made.

Again, a relation is not a subset of the set itself.

How are you defining "relation"? I don't think you are asking about the same thing we are talking about.

I mean [by a set being true or false] that set is indicated by / produced by / a fact proven from, some operation or circumstance.

It is not at all clear what you mean by this, either. I suppose a set can be defined as "all things for which [something specified] is true"; but that is not really part of what we mean by "set". The concept of "set" is a very broad generalization: any collection of anything, regardless of what criterion (or none) might lead to that particular collection. (In a similar way, the concept of "function" is a generalization of the idea of a formula or calculation, without needing any formula, just a set of inputs and corresponding outputs.)

Because the elements of X are all related to each other by being in the same set to start with. There is a relation that contains all of those elements which allows them to be in the same set -- that relation isn't a subset. If there were no relation in which the elements weren't presuppositionally related, they'd bear no relation: not even a common set in which to be situated.

This is where I think you are using the term "relation" outside of the mathematical meaning. We don't say two elements are related because there is some connection you can find between them; rather, we can identify any set of pairs of elements and call that a relation. It is a relation not because of any preexisting relationship, but just because we chose to give a name to some subset of the Cartesian product. We can apply the concept to all sorts of "presuppositional relationships", but applications are not part of the math itself, which is abstract and general.

When you say, "that relation isn't a subset", do you mean that the relation isn't a proper subset of the Cartesian product, but the entire thing? That's true of the relation "is in set X with"; but that's why the word "subset" is different from proper subset. The entire set is a subset of itself, but not proper.

 

[pka]

Thank you, is there a source I can cite for what full relations are?

This is actually an easy question to answer: any set theory textbook in your library or on the used book market.
Here are two of my favorites.
1) NAIVE SET THEORY by Paul Halmos. It is often called "The working mathematician's handbook"
In it sections 6-8, are in order: ORDERED PAIRS, RELATIONS, & FUNCTIONS.

2) There are many many DISCRETE MATHEMATICS textbooks, anyone of which should serve you well.
One that I really like is by James Anderson: DISCRETE MATHEMATICS with Combinatorics.
$15.00 on the used marked.

 

[SenatusPopulusqueRomanus]

I was looking for a source of your claim that "set theorists say that this indicates a subset that contains those relations". I think you are probably misunderstanding a number of things, so it will be helpful to see what was actually said that you are asking about.

The source is the Wikipedia pages on relations and ordered pairs, for example “A binary relation between sets A and B is a subset of A × B.”

Your wording here makes no sense in terms of the actual definition of a relation; a relation is a subset of the Cartesian product of two sets, by definition -- not a subset of the set(s) being related.

Then this is a part of the answer I came here to find.

If no one actually says this,

I didn’t say that no one says this, I said that I don’t think there is a source within mathematics for it. But if I had, that wouldn’t entail the rest of your assertion either.

If no one actually says this, then it would appear to be a figment of your own imagination, and there is nothing to be said about it mathematically.

No it does not appear that way, and the way you think it appears isn’t even sound reasoning.
If a claim has no currently existing source, then it’s just a figment of imagination?
And if a claim has no currently existing source, there's nothing to be said of it mathematically?

Every single mathematical claim once had no source, and every single source within mathematics began as a figment of someone’s imagination. A claim having no source doesn't entail either of these. So I don’t really know what to say to this assertion other than that it’s not correct. It entails a question-begging fallacy with regards to how mathematical knowledge is derived.

Aside from this, I gave exact reasons in other comments here why I make the claim, so it’s unfair discourse for you to dismiss it as “a figment of my imagination”.

there is nothing to be said about it mathematically

This is where I think you are using the term "relation" outside of the mathematical meaning.

Why does any element of one set being related to any element of another set (or the same set) have nothing to be said about it mathematically?

It is not at all clear what you mean by this, either.

It's not clear why you think this, either. Without an explanation I have no reason to accept that what I said isn’t clear.

I suppose a set can be defined as "all things for which [something specified] is true"

Then you can understand this to mean what I meant when I asked “how a set can be proven to be true”.

We don't say two elements are related because there is some connection you can find between them; rather, we can identify any set of pairs of elements and call that a relation.

I don’t see why these two things are mutually exclusive. In fact, I’m struggling to see why they might be different. Pairings between elements are connections.

It is a relation not because of any preexisting relationship, but just because we chose to give a name to some subset of the Cartesian product.

As above, I don’t see the mutual exclusion between those two things. Our ability to choose is a preexisting relationship with regards to mathematical relations. It holds for literally anything mathematical you can conceive of.

When you say, "that relation isn't a subset", do you mean that the relation isn't a proper subset of the Cartesian product, but the entire thing? That's true of the relation "is in set X with"; but that's why the word "subset" is different from proper subset. The entire set is a subset of itself, but not proper.

I don’t see this as having any impact on my assertion. The many-to-many / any-to-any relation I referred to earlier isn’t a subset, nor a proper subset, of the set(s) it relates.

 

[SenatusPopulusqueRomanus]

1) NAIVE SET THEORY by Paul Halmos. It is often called "The working mathematician's handbook"
In it sections 6-8, are in order: ORDERED PAIRS, RELATIONS, & FUNCTIONS.

Thank you, I see on page 27 that it has the rule I was trying to find.

 

[Dr.Peterson]

The source is the Wikipedia pages on relations and ordered pairs, for example “A binary relation between sets A and B is a subset of A × B.”

But what you said was,

I'm looking for a rule to cite about how a mapping or relation between elements of two sets implies a relation originating from the superset of those sets of elements.
I understand that typically, if elements of two different sets are paired by a relation or mapping, set theorists say that this indicates a subset that contains those relations.

All I was trying to say is that what you said is not expressed correctly (because you are using terminology incorrectly), and to ask you to show more clearly what you are asking about. I said,

Your wording here makes no sense in terms of the actual definition of a relation; a relation is a subset of the Cartesian product of two sets, by definition -- not a subset of the set(s) being related.

So it looks like you've recognized the difference, and understand that what you said was incorrect. We're done.

Thank you, I see on page 27 that it has the rule I was trying to find.

I don't find anything on that page of Halmos (assuming I have the same edition) that appears to be "the rule" you might be referring to; could you quote it, so we can see what you had in mind, and understand you better?

 

[SenatusPopulusqueRomanus]

But what you said was,

I'm looking for a rule to cite about how a mapping or relation between elements of two sets implies a relation originating from the superset of those sets

Yes, because I'm wanting to verify whether there is or isn't any proof in mathematics whether a relation is ever a superset of such (because what I am seeking to prove might already be proven). If there isn't, I only have those where relations are expressed as a kind of subset (or equal to the Cartesian product, as you've all helped me realize here) at my disposal.

All I was trying to say is that what you said is not expressed correctly (because you are using terminology incorrectly)

Yes, thank you.

I don't find anything on that page of Halmos (assuming I have the same edition) that appears to be "the rule" you might be referring to; could you quote it, so we can see what you had in mind, and understand you better?

In the penultimate paragraph of that page, it says "If R = X x Y". The fact that it can be equal to the Cartesian product is a useful basis for me to proceed from.

 

[Dr.Peterson]

Yes, because I'm wanting to verify whether there is or isn't any proof in mathematics whether a relation is ever a superset of such (because what I am seeking to prove might already be proven). If there isn't, I only have those where relations are expressed as a kind of subset (or equal to the Cartesian product, as you've all helped me realize here) at my disposal.

You still need to be more precise about what you mean. What is "such"?

A relation is, by definition, any subset (proper or improper) of the Cartesian product. Therefore it can't be a (proper) superset of the Cartesian product. What are you suggesting it can be a superset of?

You can, of course, define a new relation that is a subset of the given relation, and then the given relation will be a superset of the new, smaller relation. For example, the relation between people, "is a parent of", is a superset of the relation "is the father of", because it includes additional pairs (mothers and children).

Is that what you have in mind?

In the penultimate paragraph of that page, it says "If R = X x Y". The fact that it can be equal to the Cartesian product is a useful basis for me to proceed from.

I see that; that means that a relation doesn't have to be a proper subset of X x Y, but can be the entire Cartesian product (not, however, a superset of it).

Of course, higher up on the page, he says,

The least exciting relation is the empty one. ... Another dull example is the Cartesian product of any two sets X and Y.​

So he doesn't consider the case you are asking about to be very interesting! It relates every element of one set to every element of the other, making no restriction. (It is the restriction that makes a relation interesting; if all numbers were equal to one another, for example, there would be very little math to do.)

Which leaves us with the question we've had all along: Why are you interested in this case, or in supersets? If you gave an example of what you want to do, it would go a long way toward making this a profitable discussion.