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

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!

**Figure.** Basic Set Theory.

There are a few unary operators.

- \(S = \{\} = \varnothing\)
- A set S with no elements is an empty set.
- ∅ = ∅ = ∅ = ∅.
- \(\emptyset\) = \emptyset.
- \(\varnothing\) = \varnothing.

- \(powerset(S)\)
- The power set of set S is a set whose members are all possible subsets of set S.
- EG: powerset({1, 2}) = {{}, {1}, {2}, {1, 2}}.

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.

- \(e \in S\)
- e is an element of set S.
- EG: 1 ∈ {1, 2, 3}.
- ∈ = ∈ = ∈ = ∈.
- \(\in\) = \in.

- \(e \notin S\)
- e is not an element of set S.
- EG: 1 ∉ {4, 5, 6}.
- ∉ = ∉ = ∉ = ∉
- \(\notin\) = \notin = \not\in.

- \(S \ni e\)
- Set S contains as a member e.
- EG: {1, 2, 3} ∋ 1.
- ∋ = ∋ = ∋ = ∋.
- \(\ni\) = \ni.

- \(S \not\ni e\)
- Set S does not contain as a member e.
- EG: {1, 2, 3} ∋ 4.
- \(\not\ni\) = \not\ni.

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

- \(A = B\)
- Set A is equal to set B.
- EG: A = A.
- EG: {1, 2} = {1, 2}.

- \(A \ne B\)
- Set A is not equal to set B.
- EG: {1} ≠ {1, 2}.
- EG: {1, 2} ≠ {1}.
- EG: {1, 2} ≠ {3, 4}.
- ≠ = ≠ = ≠ = ≠.
- \(\ne\) = \ne = \neq
- !=.
- <>.

- \(A \subset B\)
- Set A is a subset of set B.
- Analagous to A < B.
- EG: {1} ⊂ {1, 2}.
- ⊂ = ⊂ = ⊂ = ⊂.
- \(\subset\) = \subset.

- \(B \not\subset A\)
- Set B is not a subset of set A.
- Analagous to A \(\nless\) B.
- EG: {1, 2} ⊄ {3, 4}.
- EG: {1, 2} ⊄ {2, 3}.
- ⊄ = ⊄ = ⊄ = ⊄.
- \(\not\subset\) = \not\subset.
- !⊂.

- \(A \subseteq B\)
- Set A is a subset of or equal to set B.
- Analagous to A ≤ B.
- EG: {1} ⊆ {1, 2}.
- EG: {1, 2} ⊆ {1, 2}.
- ⊆ = ⊆ = ⊆ = ⊆.
- \(\subseteq\) = \subseteq.
- ⊂=.

- \(B \supset A\)
- Set B is a superset of set A.
- Analagous to B > A.
- EG: {1, 2} ⊃ {1}.
- ⊃ = ⊃ = ⊃ = ⊃.
- \(\supset\) = \supset.

- \(A \not\supset B\)
- Set B is not a superset of set A.
- Analagous to B \(\ngtr\) A.
- EG: {1} ⊅ {1, 2}.
- ⊅ = ⊃ = ⊃ = ⊃.
- \(\not\supset\) = \not\supset.
- !⊃.

- \(B \supseteq A\)
- Set B is a superset of or equal to set A.
- Analagous to B ≥ A.
- EG: {1, 2} ⊃ {1}.
- EG: {1, 2} ⊇ {1, 2}.
- ⊇ = ⊇ = ⊇ = ⊇.
- \(\supseteq\) = \supseteq.
- ⊃=.

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

There are binary operators between sets.

- \(A \cup B\)
- Set A union set B.
- The set of all elements in either A or B.
- Analagous to A | B.
- EG: {1, 2, 3} ∪ {2, 3, 4} = {1, 2, 3, 4}.
- ∪ = ∪ = ∪ = ∪.
- \(\cup\) = \cup.

- \(A \cap B\)
- Set A intersection set B.
- The set of elements in both A and B.
- Analagous to A & B.
- EG: {1, 2, 3} ∩ {2, 3, 4} = {2, 3}.
- If A ∩ B = ∅, then the sets A and B are
**disjointed**, i.e. they have no common elements. - EG: {1, 2} ∩ {3, 4} = ∅.
- ∩ = ∩ = ∩ = ∩
- \(\cap\) = \cap.

- \(A \setminus B\)
- Set A set difference set B.
- The set A without any elements in set B.
- Analagous to A - B.
- EG: {1, 2, 3,} \ {2, 3, 4} = {1}.
- This operation is not commutative, i.e usually (A \ B) ≠ (B \ A).
- EG: {1, 2} \ {2, 3} = {1}.
- EG: {2, 3} \ {1, 2} = {3}.
- If (A ⊆ B) is true, then (B \ A) is called "the complement of A in B". The complement of A is sometimes denoted as A
^{c}, especially if B is the universal set. - EG: B \ A = A
^{c}= {1, 2, 3} \ {1} = {2, 3}. - \(\setminus\) = \setminus
- \(\smallsetminus\) = \smallsetminus

- \(A \triangle B\)
- Set A symmetric difference set B.
- The set whose members are in A or B, but not both.
- Analagous to (A | B) - (A & B).
- EG: ({1, 2, 3} Δ {2, 3, 4}) = {1, 4}.
- Δ = Δ = Δ = Δ.
- △ = △ = △ = geometric shape triangle.
- \(\triangle\) = \triangle = \bigtriangleup.
- \(\vartriangle\) = \vartriangle.
- \(\ominus\) = \ominus. Meh.

- A × B
- Set A cartesian product set B.
- The set of all possible ordered pairs between the elements of set A and set B.
- This operation is not commutative, i.e usually (A × B) ≠ (B × A).
- EG: {1, 2} × {3, 4} = {(1, 3), (1, 4), (2, 3), (2, 4)}.
- EG: {3, 4} × {1, 2} = {(3, 1), (3, 2), (4, 1), (4, 2)}.
- × = × = × = ×.
- \(\times\) = \times.
- x

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}.

- Set: A collection of unordered, unique, and finite elements. EG: A = {1, 2, 3} = {3, 1, 2}.
- Sequence: A collection of ordered, not-necessarily-unique, and not-necessarily finite elements. EG: B = (1, 1, 2, 3, ...).
- Tuple: A collection of ordered, not-necessarily-unique, and finite elements. EG: C = [1, 1, 2, 3].

- That leaves: A collection of unordered, not-necessarily-unique, and not-necessarily finite elements. EG: D = <1, 3, 1, 7, ..., >. Why are these collections unnamed?

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 that lead to off-site pages about set theory.

Wikipedia:

- Set theory
- Group theory
- Euler diagram
- Venn diagram
- Syllogism
- Prior Analytics. By Aristotle.

Miscellany:

- "Diagrams" [http://plato.stanford.edu/entries/diagrams/].
- "Fun with Venn and Euler Diagrams" [http://www.mentalfloss.com/blogs/archives/12219].
- "Venn Vs Euler: The Diagrams" [http://blog.stevemould.com/venn-vs-euler-diagrams/].

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