Set

A **set** is the mathematical model for a collection of different things; a set contains *elements* (so a collection of elements).

We often think about sets as lying in some *universal set* (often denoted

Ω

{\displaystyle \Omega }

,

S

{\displaystyle S}

,

U

{\displaystyle \mathbb {U} }

, or

U

{\displaystyle U}

), which can be thought of as a collection that contains all the entities one wishes to consider in a given situation.

## Notation

A set is usually denoted by a capital letter, e.g.

A

{\displaystyle A}

. If an element

a

{\displaystyle a}

 is in the set we write

a
∈
A

{\displaystyle a\in A}

 or

a
∉
A

{\displaystyle a\notin A}

 if not in the set.

[Q: WOULD IT BE RIGHT TO SAY THAT ELEMENT, SUBSET, AND SET EQUALITY ARE BINARY RELATIONS; EMPTY SET, DISJOINT AND CLASS ARE DEFINITIONS, AND COMPLEMENT, UNION, ETC. ARE BINARY OPERATIONS?]

There are generally three methods to denote (and possibly define sets).

1. **Roster** (a.k.a. **enumeration**) **notation**: specifies a set by a list of elements in curly brackets, e.g.

{
a
,
b
,
c
}

{\displaystyle \{a,b,c\}}

. If there are many elements with an implicit pattern, often denoted using triple colons as an abbreviation as e.g.

{

a

1

,

a

2

,
.
.
.
,

a

N

}

{\displaystyle \{a\_{1},a\_{2},...,a\_{N}\}}

, or for infinite sets by

{

a

1

,

a

2

,
.
.
.
}

{\displaystyle \{a\_{1},a\_{2},...\}}

.
2. **Semantic definition**: specifies a set by a 'rule' that all elements satisfy. E.g. let

A

{\displaystyle A}

 be the set of all even numbers.
3. **Set-builder notation**: specifies a set as a selection from a larger set. E.g.

# F
{
x
∈

R

∣

x is an integer

}

{\displaystyle F=\{x\in \mathbb {R} \mid {\text{x is an integer}}\}}

.

## Subset and Superset

* A

{\displaystyle A}

 is a subset of

B

{\displaystyle B}

 if a every element in

A

{\displaystyle A}

 is also in

B

{\displaystyle B}

,

A
⊆
B

⟺

∀
x
∈
A
,
x
∈
B

{\displaystyle A\subseteq B\iff \forall x\in A,x\in B}
* A

{\displaystyle A}

 is a superset of

B

{\displaystyle B}

 if every element in

B

{\displaystyle B}

 is also in

A

{\displaystyle A}

,

A
⊇
B

⟺

∀
x
∈
B
,
x
∈
A

{\displaystyle A\supseteq B\iff \forall x\in B,x\in A}

Hence, if

A

{\displaystyle A}

 is a subset of

B

{\displaystyle B}

, then

B

{\displaystyle B}

 is a superset of

A

{\displaystyle A}

.

# A
B

{\displaystyle A=B}

, if they contain exactly the same elements. That is, if

∀
x
∈
A
,

x
∈
B

{\displaystyle \forall x\in A,~x\in B}

 and

∀
y
∈
B
,

y
∈
A

{\displaystyle \forall y\in B,~y\in A}

.

* Note: it follows from this definition that

{
2
,
4
,
6
# }
{
2
,
6
,
4
}

{\displaystyle \{2,4,6\}=\{2,6,4\}}

. Hence, when writing a set in roster notation, the order does not matter.
* Note: it follows from this definition that

{
2
,
4
,
4
# }
{
2
,
4
}

{\displaystyle \{2,4,4\}=\{2,4\}}

. Hence, when writing a set in roster notation, repeated elements do not change the set. E.g. the set of letters in the word *Mississippi* is equal to the set

{
M
,
i
,
s
,
p
}

{\displaystyle \{M,i,s,p\}}

.
* Hence, it follows from the definition of set equality that

{
a
,
1
,
1
# }
{
1
,
a
}

{\displaystyle \{a,1,1\}=\{1,a\}}

.
* Note: a theorem follows from the definitions of subset and set equality :

# A
B

⟺

A
⊆
B

and

B
⊆
A

{\displaystyle A=B\iff A\subseteq B~{\text{and}}~B\subseteq A}

. (note, this is a theorem as it is not a definition itself, but follows from two definitions)

Alternatively, may want to rewrite this part as; properties of set equality and add their proofs from the definition of set equality.

* {
x
# }
{
x
,
x
}

{\displaystyle \{x\}=\{x,x\}}
* {
x
,
y
# }
{
y
,
x
}

{\displaystyle \{x,y\}=\{y,x\}}
* {
x
# }
{
y
,
z
}

⟺

# y
z

{\displaystyle \{x\}=\{y,z\}\iff x=y=z}

## Empty Set

The **empty set** is a set with no elements in it, and is denoted as

# ∅
{
}

{\displaystyle \varnothing =\{\}}

.

## Class/Set of Sets

A set whose elements are sets (a set of sets) is called a **class**, **family**, **collection**, or **set system**. Often, a family is denoted with a calligraphic letter like

A

1

,

A

2

∈

A

{\displaystyle A\_{1},A\_{2}\in {\mathcal {A}}}

 where

A

1

,

A

2

{\displaystyle A\_{1},A\_{2}}

 are sets (and

A

⊆

2

Ω

{\displaystyle {\mathcal {A}}\subseteq 2^{\Omega }}

 where

Ω

{\displaystyle \Omega }

 is the universal set).

Some special families/classes of sets, when addition structure is added (see Space (math)), have special names, e.g. pi-systems, lambda-systems, algebras, sigma-algebras, and topologies.

## Disjoint Sets

* **Disjoint**: If the intersection of two sets is empty

A
∩
# B
∅

{\displaystyle A\cap B=\varnothing }

, then the two sets are disjoint*.*
* **Pairwise Disjoint**: a family of sets is pairwise disjoint if any two sets in it are disjoint; if for any

A

i

,

A

j

∈
(

A

i

)

i
∈
I

{\displaystyle A\_{i},A\_{j}\in (A\_{i})\_{i\in I}}

whenever

i
≠
j

{\displaystyle i\neq j}

,

A

i

∩

A

j

=
∅

{\displaystyle A\_{i}\cap A\_{j}=\varnothing }

.

Note that if two sets are disjoint, then

A
∖
# B
A

{\displaystyle A\setminus B=A}

 and

A
△
# B
A
∪
B

{\displaystyle A\triangle B=A\cup B}

.

## Power Set

The **power set** of a set

A

{\displaystyle A}

 is the set of all subsets of

A

{\displaystyle A}

, including the empty set and

A

{\displaystyle A}

 itself, denoted by

2

A

{\displaystyle 2^{A}}

. Hence, the power set is a family/class.

The power set can not be empty, by definition it contains at least two elements,

∅

{\displaystyle \varnothing }

 and

A

{\displaystyle A}

.

## Basic Set Operations

For more results, e.g. De Morgan’s laws, see Algebra of Sets. Assuming the sets that we study live in some larger set

Ω

{\displaystyle \Omega }

, the following basic operations are defined:

* **Complement**:

A

c

=
{
x
∈
Ω
∣
x
∉
A
}

{\displaystyle A^{c}=\{x\in \Omega \mid x\notin A\}}

, the set of elements not in a set.
* **Intersection**:

A
∩
# B
{
x
∈
Ω
∣
x
∈
A

 and

x
∈
B
}

{\displaystyle A\cap B=\{x\in \Omega \mid x\in A{\text{ and }}x\in B\}}

, the set of elements that are both in set

A

{\displaystyle A}

 and set

B

{\displaystyle B}

.
* **Union**:

A
∪
B
:=
{
x
∈
Ω
∣
x
∈
A

 or

x
∈
B
}

{\displaystyle A\cup B:=\{x\in \Omega \mid x\in A{\text{ or }}x\in B\}}

, the set of elements in

A

{\displaystyle A}

,

B

{\displaystyle B}

, or both.
* **Difference**:

A
∖
B
:=
{
x
∈
A
∣
x
∉
B
# }
A
∩

B

c

{\displaystyle A\setminus B:=\{x\in A\mid x\notin B\}=A\cap B^{c}}

, the elements in A that are not in B.
* **Symmetric Difference**:

A
△
B
:=
(
A
∪
B
)
∖
(
A
∩
B
)

{\displaystyle A\triangle B:=(A\cup B)\setminus (A\cap B)}

, the elements in

A

{\displaystyle A}

 or

B

{\displaystyle B}

, but not in both.
* **Cartesian Product**:

A
×
B
:=
{
(
x
,
y
)
∣
x
∈
A

 and

y
∈
B
}

{\displaystyle A\times B:=\{(x,y)\mid x\in A{\text{ and }}y\in B\}}

, this is a way of defining ordered pairs/sequences/lists from sets. [ADD AND LINK TO ARTICLE WITH DETAILS ON OPERATIONS ON CARTESIAN PRODUCT].

## Important Sets of Numbers

Some sets are referred to more often, and have been given their own notation. These include:

* N

{\displaystyle \mathbb {N} }

:

{
1
,
2
,
3
,
.
.
.
}

{\displaystyle \{1,2,3,...\}}

, the 'natural' or 'counting' numbers (conventions differ on whether zero is included).
* Z

{\displaystyle \mathbb {Z} }

:

{
.
.
,
−
2
,
−
1
,
0
,
1
,
2
,
.
.
.
}

{\displaystyle \{..,-2,-1,0,1,2,...\}}

, the set of integers
* Q

{\displaystyle \mathbb {Q} }

: the set of all rational numbers (that is, the set of all proper and improper fractions)
* R

{\displaystyle \mathbb {R} }

: the set of all real numbers
* C

{\displaystyle \mathbb {C} }

: the set of all complex numbers,

{
a
i
b

a
,
b
∈

R

}

{\displaystyle \{a+ib~|~a,b\in \mathbb {R} \}}
* R

n

{\displaystyle \mathbb {R} ^{n}}

: the set of all real-valued

n
×
1

{\displaystyle n\times 1}

 dimensional vectors
* C

(
A
,
B
)

{\displaystyle {\mathcal {C}}(A,B)}

: the set of all continuous functions from set

A

{\displaystyle A}

 to set

B

{\displaystyle B}

.

Clearly,

N

⊂

Z

⊂

Q

⊂

R

⊂

C

{\displaystyle \mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} }

.

1. ↑ In Latex:
```
\varnothing

```
2. ↑ Sometimes also denoted as

P
(
A
)
,

P

(
A
)
,

P

(
A
)

{\displaystyle P(A),\mathbb {P} (A),{\mathcal {P}}(A)}

, etc.
3. ↑ add link to fractions page for the different types