basics

set

A set is a collection of distinct objects which we call elements of the set. A set can have a finite number of elements or infinitely many elements.

declaration of a finite set

We may declare a finite set by listing it's elements between curly braces like so

{a, b, c, d}

{1, 2, 3, 41}

Suppose we wrote out the set

{a, b, c, d}

and then later on realized that

a = b = c = d

, then this set is simply

{a, a, a, a}

, which we define to just be

{a}

. Therefore we allow duplicates when declaring a set, but realize the set it generates has no such duplicate, so

{1, 1, 1} = {1, 1} = {1}

set builder notation

Suppose that

P (x)

represents a statement which depends on the variable

x

, then we define

{x \in S : P (x)}

to be the set of elements of

S

such that the statement

P

holds true.

Empty Set

The set with no elements is called the empty set and is denoted by

\emptyset

element of

Suppose that

a

is an element of a set

S

, then we write

a \in S

set equality

Given two sets

A, B

, we say that

A

and

B

are equal and write

A = B

when

x \in A

if and only if

x \in B

Family of Sets

We denote a set whose elements are also sets by a family of sets to avoid the unwieldy statement: "set of sets"

subset

Given the sets

A, B

, we say that

A

is a subset of

B

when for every

a \in A

a

is also in

B

. When this is the case we write

A \subseteq B

superset

Given the sets

A, B

, we say that

A

is a superset of

B

when for every

b \in B

b

is also in

A

. When this is the case we write

A \supseteq B

set equality by subsets

Suppose that

A, B

are sets such that

A \subseteq B

and

B \subseteq A

, then

A = B

We will show the sets are equal using the definition of set equality.

To show that a bi-implication is true, we have to prove both directions. So first suppose that $x \in A$ , then since $A \subseteq B$ , $x \in B$ . For the other direction suppose that $x \in B$ , then since $B \subseteq A$ then $x \in A$ . Therefore we know that $x \in A$ if and only if $x \in B$ as needed.

rationals

The integers are the set

ℚ : = {\frac{p}{q} : p, q \in ℤ, q \neq 0}

note: The symbol for rationals can be remebered as they are $ℚ$ uotients

integers

The integers are the set

ℤ : = {\dots, - 3, - 2, - 1, 0, 1, 2, 3, \dots}

note: The symbol for integers can be remembered as the word for number in German is $ℤ$ ahlen

Natural Numbers 0

We define

ℕ_{0} : = {0, 1, 2, 3, 4, \dots}

Finite Integer Set

Suppose that

a, b \in Z

, then the notation

{a, \dots b}

is a set

X

such that for any

x \in Z

a \leq x \leq b

then

x \in X

Empty Integer Set

Suppose that

a, b \in Z

such that

a > b

then

{a, \dots b} = \emptyset

{a, \dots b}

is defined as the set

X

such that

x \in X

iff

x \in Z

and

a \leq x \leq b

but this is impossible for if there were any such

x

then by transitivity we have that

a \leq b

which is impossible.

Natural Numbers 1

We define

ℕ_{1} : = {1, 2, 3, 4, \dots}

Natural Numbers up to n

Let

n \in ℕ_{1}

, then we define

[n] : = {1, 2, \dots, n}

power set

Given a set

X

we define the power set of subsets

X

P (X) : = {S : S \subseteq X}

🏗️ ΘρϵηΠατπ🚧 (under construction)

🏗️ $Θ ρ ϵ η Π α τ π$ 🚧 (under construction)