Exploring set theory, a branch of math that studies sets.

Intro

Set theory is deals with elements in sets. Sets are just about everywhere, so most of us intuitively deal with set theory all the time!

Before I get to sets, let me start with the elements in sets. An element can be just about anything (numbers, nouns, verbs, sets, etc). An element may be referred to as a member, item, object, thing, etc. An element is often represented as a lower case letter. EG: a = 1.

A set is a collection of unordered, unique, and finite elements. It is often represented as an upper case letter, and a literal set is often bounded with curly brackets. EG: A = {1, 2, 3} = {3, 1, 2}.

Now that we have elements and sets, we can start on set theory!

Basic set theory
Figure. Basic Set Theory.

Unary Relations

There are a few unary operators.

Binary Relations between e and S

There are binary relations between an element and a set, i.e. set membership or membership relations. Since a set can also be considered as an element, these relations can also be used between sets.

Binary Relations between A and B

There are binary relations between sets, i.e. set inclusion or subset relations.

There are others like \(\equiv, \not\equiv,\nsubseteq, \subsetneq, \nsupseteq, \supsetneq, \) etc., but I think you get the idea.

Binary Operators between A and B

There are binary operators between sets.

Euler and Venn Diagrams

Historically Euler diagrams predate Venn diagrams. In effect, Euler diagrams are a subset of Venn diagrams. Venn diagrams show all possible logical relations between a finite collection of sets, whereas a Euler diagram often leaves out relations that yield empty sets.

There are a several collection related concept similar to a set, so I will make the distinctions here. Collections are often represented with an upper case letter. EG: A = {1, 2, 3}.

My plan is to show collections based on ordered, uniqueness, and finiteness as a truth table, then as a Venn diagram, then as a Euler diagram. The Euler will match the set, sequence, and tuple divisions.

Links

Links that lead to off-site pages about set theory.

Wikipedia:

Miscellany:

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