Set Theory by George Hernandez

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

Intro

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

There are special characters used in set theory and Web folks have been nice enough to provided Character Entity References (CERs) for them. Here they are ordered by Unicode point:

=
\. To represent set difference.
× = × = × = ×. "multiplication sign". To represent cartesian product.
Δ = Δ = Δ = Δ. "greek capital letter delta". Sometimes -.
≠ = ≠ = ≠ = ≠. "not equal to". Sometimes != or <>.
∅ = ∅ = ∅ = ∅. "empty set".
∈ = ∈ = ∈ = ∈. "element of".
∉ = ∉ = ∉ = ∉. "not an element of".
∋ = ∋ = ∋ = &ni;. "contains as member".
∩ = ∩ = ∩ = ∩. "intersection".
∪ = ∪ = ∪ = ∪. "union".
⊂ = ⊂ = ⊂ = ⊂. "subset of".
⊃ = ⊃ = ⊃ = ⊃. "superset of".
⊄ = ⊄ = ⊄ = &nsub;. "not a subset of". Sometimes !⊂.
⊆ = ⊆ = ⊆ = &sube;. "subset of or equal to". Sometimes ⊂=.
⊇ = ⊇ = ⊇ = &supe;. "superset of or equal to". Sometimes ⊃=.

Before I get to sets, let me start with the elements in sets. An item can be just about anything (numbers, nouns, verbs, sets, etc). An item 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. EG: A = {1, 2, 3} = {3, 1, 2}.

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

There are a few unary operators:

S = {} = ∅. A set S with no elements is an empty set.
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 operators between an element and a set, i.e. set membership:

e ∈ S. An element e is an element of set S. EG: 1 ∈ {1, 2, 3}.
S ∋ e. A set S contains as a member the element e . EG: {1, 2, 3} ∋ 1.
e ∉ S. An element e is not an element of set S. EG: 1 ∉ {4, 5, 6}.

There are binary comparisons between sets, i.e. set inclusion:

A = B. Set A is equal to set B. Set A has exactly the same elements as set B. EG: {1, 2} = {1, 2}. EG: A = A.
A ≠ B. Set A is not equal to set B. Set A does not have exactly the same elements as set B. EG: {1} ≠ {1, 2}. EG: {1, 2} ≠ {1}. EG: {1, 2} ≠ {3, 4}.
A ⊄ B. Set A is not a subset of set B. Not all the elements of set A are in set B. EG: {1, 2} ⊄ {3, 4}. EG: {1, 2} ⊄ {2, 3}.
A ⊂ B. Set A is a subset of set B. All the elements of set A are in set B but set B has other elements. Analagous to a < b. EG: {1} ⊂ {1, 2}.
A ⊆ B. Set A is a subset of or equal to set B. All the elements of set A are in set B and possibly the same as set B. Analagous to a <= b. EG: {1} ⊆ {1, 2}. EG: {1, 2} ⊆ {1, 2}.
B ⊃ A. Set B is a superset of set A. Set B has all the elements in set A, but set B has other elements. Analagous to b > a. EG: {1, 2} ⊃ {1}.
B ⊇ A. Set B is a superset of or equal to set A. Set B has all the elements in set A, but set B may have other elements. Analagous to b > a. EG: {1, 2} ⊃ {1}. EG: {1, 2} ⊇ {1, 2}.

There are binary operators/relations between sets:

A ∪ B. Set A union set B. A set of all elements in A, B, or both. EG: {1, 2, 3} ∪ {2, 3, 4} = {1, 2, 3, 4}.
A ∩ B. Set A intersection set B. The set of elements both A and 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} = ∅.
A \ B. Set A set difference set B. The set A without any elements in set B. This operation is not commutative, i.e usually (A \ B) ≠ (B \ A). Analagous to a - b. EG: {1, 2, 3,} \ {2, 3, 4} = {1}. 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"; EG: {1, 2, 3} \ {1} = {2, 3}.
A Δ B. Set A symmetric difference set B. The set of elements whose members are in A or B, but not both. EG: ({1, 2, 3} Δ {2, 3, 4}) = {1, 4}.
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)}.

Basic set theory

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.

Links

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

Wikipedia:

Miscellany:

Page Modified: (Hand noted: 2011-09-29 15:23:14Z) (Auto noted: 2011-09-30 20:30:44Z)