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Mathematical Sets by Wiki.

Subsets by Wiki.

 

Set Operations (A Wiki Rip)

There are ways to construct new sets from existing ones. Two sets can be “added” together. The union of A and B, denoted by A ∪ B, is the set of all things which are members of eitherA or B.

 

The union of A and B, or A ∪ B

Examples:

  • {1, 2} ∪ {red, white} = {1, 2, red, white}.
  • {1, 2, green} ∪ {red, white, green} = {1, 2, red, white, green}.
  • {1, 2} ∪ {1, 2} = {1, 2}.

Some basic properties of unions are:

  • A ∪ B = B ∪ A.
  • A ∪ (B ∪ C) = (A ∪ B) ∪ C.
  • A ⊆ (A ∪ B).
  • A ∪ A = A.
  • A ∪ ∅ = A.
  • A ⊆ B if and only if A ∪ B = B.

[edit]Intersections

A new set can also be constructed by determining which members two sets have “in common”. The intersection of A and B, denoted by A ∩ B, is the set of all things which are members of both A and B. If A ∩ B = ∅, then A and B are said to be disjoint.

 

The intersection of A and B, orA ∩ B.

Examples:

  • {1, 2} ∩ {red, white} = ∅.
  • {1, 2, green} ∩ {red, white, green} = {green}.
  • {1, 2} ∩ {1, 2} = {1, 2}.

Some basic properties of intersections:

  • A ∩ B = B ∩ A.
  • A ∩ (B ∩ C) = (A ∩ B) ∩ C.
  • A ∩ B ⊆ A.
  • A ∩ A = A.
  • A ∩ ∅ = ∅.
  • A ⊆ B if and only if A ∩ B = A.

[edit]Complements

Two sets can also be “subtracted”. The relative complement of A in B (also called the set theoretic difference of B and A), denoted by B \A, (or B −A) is the set of all elements which are members of B, but not members of A. Note that it is valid to “subtract” members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so has no effect.

In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A′.

 

The relative complement
of A in B.

 

The complement of A in U.

Examples:

  • {1, 2} \ {red, white} = {1, 2}.
  • {1, 2, green} \ {red, white, green} = {1, 2}.
  • {1, 2} \ {1, 2} = ∅.
  • {1, 2, 3, 4} \ {1, 3} = {2, 4}.
  • If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then the complement of E in U is O, or equivalently, E′ = O.

Some basic properties of complements:

  • A ∪ A′ = U.
  • A ∩ A′ = ∅.
  • (A′)′ = A.
  • A \ A = ∅.
  • U′ = ∅ and ∅′ = U.
  • A \ B = A ∩ B.

Cartesian product

A new set can be constructed by associating every element of one set with every element of another set. The Cartesian product of two sets A and B, denoted by A × B is the set of all ordered pairs (ab) such that a is a member of A and b is a member of B.

Examples:

  • {1, 2} × {red, white} = {(1, red), (1, white), (2, red), (2, white)}.
  • {1, 2, green} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green), (green, red), (green, white), (green, green)}.
  • {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.

Some basic properties of cartesian products:

  • A × ∅ = ∅.
  • A × (B ∪ C) = (A × B) ∪ (A × C).
  • (A ∪ B) × C = (A × C) ∪ (B × C).

Let A and B be finite sets. Then

  • | A × B | = | B × A | = | A | × | B |.
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A visual of a Cantor Set

A visual of a Cantor Set

Step One: Gentle intro

“Are rational numbers countable?”

Step Two: The Hotel Infinity

“Related to Cantor’s argument that rational numbers are countable?”

Step Three: Self Similarity

“Can you write out details of constructing the Cantor Set?”