Partial Order Relation/Equivalence Relation between two sets of different size or elements

Nov 2019
1
0
Pakistan
This question is basically about the following four properties of sets,

- Reflexive Relation
- Symmetric Relation
- Antisymmetric Relation
- Transitive Relation


I know how to find all these relations when a single set is being considered.
However, what if the sets are different? For example, consider the following two sets of same size,

A = {1, 2}
B = {2, 3}
or A = {1, 2}
B = {3, 4} (totally different elements from first set)

Then their possible relations will be all the subsets of their Cartisean Product.

Can anyone of those relations be reflexive, symmetric, antisymmetric, transitive? If I check from Matrix Representation of those relations, it shows that those relations do have all these properties. However, from definition, it tells you that not a single relation has any of these properties... Can anyone verify this please?
Also, if they do have all these properties, then what if we use sets of different sizes then? Will they also satisfy all these properties?

Thank you in advance.
 

Jhevon

MHF Helper
Feb 2007
11,681
4,225
New York, USA
As far as I know, those terms refer to a relation on a single set, and maybe a Cartesian product of a set with itself.

If we're talking about two sets with no elements in common, I don't think "reflexive", etc., would be a thing. We'd discuss other properties, such as those of a function.
 
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Jun 2013
1,112
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Lebanon
A subset of the Cartesian product \(\displaystyle A\times B\) is a binary relation over the sets \(\displaystyle A\) and \(\displaystyle B\)

a binary relation can be represented by matrices

the properties reflexive, symmetric, transitive , ... are defined when \(\displaystyle A=B\)

reference: Binary