learnxinyminutes-docs/set-theory.html.markdown

category	name	contributors
Algorithms & Data Structures	Set theory

The set theory is a study for sets, their operations, and their properties. It is the basis of the whole mathematical system.

• A set is a collection of definite distinct items.

Basic operators

These operators don't require a lot of text to describe.

• ∨ means or.
• ∧ means and.
• , separates the filters that determine the items in the set.

A brief history of the set theory

Naive set theory

• Cantor invented the naive set theory.
• It has lots of paradoxes and initiated the third mathematical crisis.

Axiomatic set theory

• It uses axioms to define the set theory.
• It prevents paradoxes from happening.

Built-in sets

• ∅, the set of no items.
• N, the set of all natural numbers. {0,1,2,3,…}
• Z, the set of all integers. {…,-2,-1,0,1,2,…}
• Q, the set of all rational numbers.
• R, the set of all real numbers.

The empty set

• The set containing no items is called the empty set. Representation: ∅
• The empty set can be described as ∅ = {x|x ≠ x}
• The empty set is always unique.
• The empty set is the subset of all sets.

A = {x|x∈N,x < 0}
A = ∅
∅ = {}              (Sometimes)

|∅|   = 0
|{∅}| = 1

Representing sets

Enumeration

• List all items of the set, e.g. A = {a,b,c,d}
• List some of the items of the set. Ignored items are represented with …. E.g. B = {2,4,6,8,10,…}

Description

• Describes the features of all items in the set. Syntax: {body|condtion}

A = {x|x is a vowel}
B = {x|x ∈ N, x < 10l}
C = {x|x = 2k, k ∈ N}
C = {2x|x ∈ N}

Relations between sets

Belongs to

• If the value a is one of the items of the set A, a belongs to A. Representation: a∈A
• If the value a is not one of the items of the set A, a does not belong to A. Representation: a∉A

Equals

• If all items in a set are exactly the same to another set, they are equal. Representation: a=b
• Items in a set are not order sensitive. {1,2,3,4}={2,3,1,4}
• Items in a set are unique. {1,2,2,3,4,3,4,2}={1,2,3,4}
• Two sets are equal if and only if all of their items are exactly equal to each other. Representation: A=B. Otherwise, they are not equal. Representation: A≠B.
• A=B if A ⊆ B and B ⊆ A

Belongs to

• If the set A contains an item x, x belongs to A (x∈A).
• Otherwise, x does not belong to A (x∉A).

Subsets

• If all items in a set B are items of set A, we say that B is a subset of A (B⊆A).
• If B is not a subset of A, the representation is B⊈A.

Proper subsets

• If B ⊆ A and B ≠ A, B is a proper subset of A (B ⊂ A). Otherwise, B is not a proper subset of A (B ⊄ A).

Set operations

Base number

• The number of items in a set is called the base number of that set. Representation: |A|
• If the base number of the set is finite, this set is a finite set.
• If the base number of the set is infinite, this set is an infinite set.

A   = {A,B,C}
|A| = 3
B   = {a,{b,c}}
|B| = 2
|∅| = 0         (it has no items)

Powerset

• Let A be any set. The set that contains all possible subsets of A is called a powerset (written as P(A)).

P(A) = {x|x ⊆ A}

|A| = N, |P(A)| = 2^N

Set operations among two sets

Union

Given two sets A and B, the union of the two sets are the items that appear in either A or B, written as A ∪ B.

A ∪ B = {x|x∈A∨x∈B}

Intersection

Given two sets A and B, the intersection of the two sets are the items that appear in both A and B, written as A ∩ B.

A ∩ B = {x|x∈A,x∈B}

Difference

Given two sets A and B, the set difference of A with B is every item in A that does not belong to B.

A \ B = {x|x∈A,x∉B}

Symmetrical difference

Given two sets A and B, the symmetrical difference is all items among A and B that doesn't appear in their intersections.

A △ B = {x|(x∈A∧x∉B)∨(x∈B∧x∉A)}

A △ B = (A \ B) ∪ (B \ A)

Cartesian product

Given two sets A and B, the cartesian product between A and B consists of a set containing all combinations of items of A and B.

A × B = { {x, y} | x ∈ A, y ∈ B }

"Generalized" operations

General union

Better known as "flattening" of a set of sets.

∪A = {x|X∈A,x∈X}
∪A={a,b,c,d,e,f}
∪B={a}
∪C=a∪{c,d}

General intersection

∩ A = A1 ∩ A2 ∩ … ∩ An