Questions about math "Relations".

[Dale10101]

In general, most simply put, a relation is a subset of a Cartesian product of two or more sets, but for the purpose of the following questions, I am thinking in terms of binary relations on the real numbers, i.e. subsets of R x R.

I read a statement claiming there are essentially four type of relations, unfortunately the article was not clear of what they might be. I am thinking (and I am not sure what the proper terms might be):

1) Relations that take and map a domain set of ordered pairs to a single element, R x R -> R, examples, addition and multiplication.
2) Relations that map an element of a set, say R, to another element of a second set (possible the same set in the cause of an "unto" relation) , example x -> 2x
3) Equivalence relations that take as a domain, the set of all relations and select those that have the properties of being reflexive, symmetric, and transitive.
4) Ordering relations that take each as a domain the pairs of R x R, for example, and determine precedence according to the properties of a partial order (reflexive, antisymmetric, and transitive), or the properties of a strict order, or perhaps properties on another type of order.

Questions:
1) Have I properly distinguished the different types of relations above (if so, are then any names distinguishing the first two in particular)?
2) Are there additional categories of relations?
3) Is an equivalence relation necessarily ordered, that is, just because two pairs can be shown to be equivalent is there a necessity that, one pair to another, there is necessarily a precedence?

Thank you.

 

[HallsofIvy]

In general, most simply put, a relation is a subset of a Cartesian product of two or more sets, but for the purpose of the following questions, I am thinking in terms of binary relations on the real numbers, i.e. subsets of R x R.

I read a statement claiming there are essentially four type of relations, unfortunately the article was not clear of what they might be. I am thinking (and I am not sure what the proper terms might be):

1) Relations that take and map a domain set of ordered pairs to a single element, R x R -> R, examples, addition and multiplication.

You said, above, "I am thinking in terms of binary relations on the real numbers, i.e. subsets of R x R." and your very first relation violates that! Yes, this is a relation that relates pairs of real numbers to single real numbers. This is a relation, a subset of "A x B" where A is itself "R x R" and B is R.

2) Relations that map an element of a set, say R, to another element of a second set (possible the same set in the cause of an "unto" relation) , example x -> 2x

Well, that is precisely the definition of "relation" so this is not a special kind of relation.

3) Equivalence relations that take as a domain, the set of all relations and select those that have the properties of being reflexive, symmetric, and transitive.

That's a peculiar way of phrasing it. An "equivalence relation" is a single relation that is reflexive, symmetric and transitive NOT a function or relation on the set of relations.

4) Ordering relations that take each as a domain the pairs of R x R, for example, and determine precedence according to the properties of a partial order (reflexive, antisymmetric, and transitive), or the properties of a strict order, or perhaps properties on another type of order.

"Anti-symmetric" is defined as "if (a, b) is in the relation then (b, a) is NOT. So an anti-symmetric relation cannot be reflexive (take b= a). You can have "non-strict" order relations that are reflexive and transitive or "strict" order relations that are reflexive and transitive. Typically, any transitive relation is called an "order" relation.

Questions:
1) Have I properly distinguished the different types of relations above (if so, are then any names distinguishing the first two in particular)?
2) Are there additional categories of relations?
3) Is an equivalence relation necessarily ordered, that is, just because two pairs can be shown to be equivalent is there a necessity that, one pair to another, there is necessarily a precedence?

I really don't know what you mean by "a precedence". Nor do I know what you men by "two pairs can be shown to be equivalent". If we have an equivalence relation, then we say that two elements making up a pair are equivalent. There is no comparison of "two pairs".

Thank you.

 

[Dale10101]

Query

Thanks for your detailed reply.

About this point:

"Anti-symmetric" is defined as "if (a, b) is in the relation then (b, a) is NOT. So an anti-symmetric relation cannot be reflexive (take b= a). You can have "non-strict" order relations that are reflexive and transitive or "strict" order relations that are reflexive and transitive. Typically, any transitive relation is called an "order" relation.

I follow your last two sentences and particularly find the last sentence valuable.

The first two sentences about the meaning of anti-symmetry confuse me. I thought the opposite, that in anti-symmetrical relations (a,b) = (b, a) only when a = b so that anti-symmetry is consistent with a reflexive relation.

From Wolfram and other sources:

A relation “less than or equal to" is a partial order on a set S if it has:

1) Reflexivity, a is less then or equal to a for all a element of S.
2) Anti-symmetry, a is less than or equal to b, and b is less than or equal to a, implies a = b.
3) Transitivity, a is less than or equal to b, and, and b is less than or equal to c, implies a is less than or equal to b.

Perhaps I am missing something, probably?

On another point:

I wrote:

3) Equivalence relations that take as a domain, the set of all relations and select those that have the properties of being reflexive, symmetric, and transitive.

You wrote:

“That's a peculiar way of phrasing it. An "equivalence relation" is a single relation that is reflexive, symmetric and transitive NOT a function or relation on the set of relations.”

What I was thinking was this. Suppose you are given a complete plot of an unknown relation on the Cartesian plane, it seems to me that you can characterize it as being an equivalence relation or not by the pattern it displays. If that is true then, taking the set of all possible relations that could have been plotted as the domain set, the equivalence relation qua its three properties would functionally answer the question true/false, equivalence relation or not.

I suppose I was not thinking in terms of the definition of an equivalence relation, but how it might be functionally be used so to speak. In that light does the idea, if not the labeling of it as an ad hoc definition, work?

Again, thanks.

More questions but I think I will reflect a little more.