I am new to both category & set theory. From what I have learned of set theory so far:
- Products between two(+) sets,
A×B, create a "grid" of elements which completely covers all combinations of elements
a ∈ A&
b ∈ B
(b,a)depending on which way round they were composed.
- Co-products between two(+) sets,
A∪B, are essentially disjoint unions. Aka, all elements of sets
Bare put into a single set,
C, and paired with a discriminator such that any overlaps (e.g.
a=b) are still kept and distinguished.
- (Bi)nary Relations are generalisations of products and co-products in that they represent distinct pairs. However, they do not guarantee every element pair, like products do, nor even that all elements in a given parent set will be used, like co-products.
Now, my question is that products and co-products have their respective symbolic representations in category theory (
A∪B respectively), but I have yet to see how to define a generic (bi)nary relation in notation beyond calling it some arbitrary name (as with any other generic set).
How do I correctly denote a set as being a binary relation between two sets,
B, in both set theory and category theory?
Thanks in advance.