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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.

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.

### 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.

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.

### 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′.

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 |. 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?”