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!

Figure. Basic Set Theory.

## Unary Relations

There are a few unary operators.

• $$S = \{\} = \varnothing$$
• A set S with no elements is an empty set.
• ∅ = &#x2205; = &#8709; = &empty;.
• $$\emptyset$$ = \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}}.

## 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.

• $$e \in S$$
• e is an element of set S.
• EG: 1 ∈ {1, 2, 3}.
• ∈ = &#x2208; = &#8712; = &isin;.
• $$\in$$ = \in.
• $$e \notin S$$
• e is not an element of set S.
• EG: 1 ∉ {4, 5, 6}.
• ∉ = &#x2209; = &#8713; = &notin;
• $$\notin$$ = \notin = \not\in.
• $$S \ni e$$
• Set S contains as a member e.
• EG: {1, 2, 3} ∋ 1.
• ∋ = &#x220B; = &#8715; = &ni;.
• $$\ni$$ = \ni.
• $$S \not\ni e$$
• Set S does not contain as a member e.
• EG: {1, 2, 3} ∋ 4.
• $$\not\ni$$ = \not\ni.

## Binary Relations between A and B

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}.
• ≠ = &#x2260; = &#8800; = &ne;.
• $$\ne$$ = \ne = \neq
• !=.
• <>.
• $$A \subset B$$
• Set A is a subset of set B.
• Analagous to A < B.
• EG: {1} ⊂ {1, 2}.
• ⊂ = &#x2282; = &#8834; = &sub;.
• $$\subset$$ = \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}.
• ⊄ = &#x2284; = &#8836; = &nsub;.
• $$\not\subset$$ = \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}.
• ⊆ = &#x2286; = &#8838; = &sube;.
• $$\subseteq$$ = \subseteq.
• ⊂=.
• $$B \supset A$$
• Set B is a superset of set A.
• Analagous to B > A.
• EG: {1, 2} ⊃ {1}.
• ⊃ = &#x2283; = &#8835; = &sup;.
• $$\supset$$ = \supset.
• $$A \not\supset B$$
• Set B is not a superset of set A.
• Analagous to B $$\ngtr$$ A.
• EG: {1} ⊅ {1, 2}.
• ⊅ = &#x2283; = &#8835; = &sup;.
• $$\not\supset$$ = \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}.
• ⊇ = &#x2287; = &#8839; = &supe;.
• $$\supseteq$$ = \supseteq.
• ⊃=.

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.

• $$A \cup B$$
• 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}.
• ∪ = &#x222A; = &#8746; = &cup;.
• $$\cup$$ = \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} = ∅.
• ∩ = &#x2229; = &#8745; = &cap;
• $$\cap$$ = \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 Ac, especially if B is the universal set.
• EG: B \ A = Ac = {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 = &#x0394; = &#916; = &Delta;.
• △ = &#9651; = &#x25B3; = 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)}.
• × = &#x00D7; = &#215; = &times;.
• $$\times$$ = \times.
• x

## 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}.

• 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.