Set Theory

Set Theory is about the collection of things. They can be considered as an example of Category Theory. ¹

There are multiple ways of defining set theory: ²

• Elementary Theory of the Category of Sets (ETCS)
• ZFC

This page will go over some common aspects of Set Theory regardless of the specific definition. ²

Sets

Properties of sets: ¹

• Sets can contain no elements, one element, or many elements.
• Elements in a set have no order among themselves.
• A set A is a subset of another set B, if B contains all of the elements in A.
• The empty set is always a subset of all sets.

Functions

Properties of functions: ¹

• Must always map every element in the input set to one element in an output set.
• Cannot map an element in an input set to multiple elements in an output set.
• There is always a function to go from the empty set to any other set.
• The only way you can have a function that goes to the empty set is to have the empty set as an input.
• If a function is bijective, that is every element in the input set is mapped to exactly one element in the output set, then the function is considered to be invertible.
• Functional composition is associative (it doesn't matter in which order you combine functions together; it just matters that the functions are still connected in the same order)

Isomorphism

Things that are function-like are also known as morphisms. In particular, functions that can be inverted are isomorphisms. When there exists an invertible function between two sets, then those two sets are isomorphic. ¹

Products

A Cartesian product of sets is a new set of every ordered combination you can make with one element from each set. It is associative. ²

All Cartesian products have a "projection". Projections extract the elements from each combination. For example, for a Cartesian product A x B = C, you can define the projections C -> A and C -> B to get back the A and B sets respectively. These projections are morphisms and are not invertible. ²

The product is the dual of the sum (See: Category Theory). ²

Sums

A sum of sets is a new set that contains all of the elements from multiple sets, with duplicate elements included. ²

Unlike products, you cannot go back to the original sets after combining them. This is because the sum C of A + B contains elements from A which cannot be mapped back to B and elements from B which cannot be mapped back to A. ²

The sum is the dual of the product, and is also known as the coproduct (See: Category Theory). ²