# Set Theory

A Set is a collection of well defined objects.

The objects that make up a set are known as members of the set.

Example consider a set:

A = (1,2,3)

1,2 and 3 are objects of Set A

\(1 \in A\)

\(4 \notin A\)

__Rules of Set:__

Rule 1: Given an object o, and a set S

Either o \(\in\) S or

o \(\notin\) S

Rule 2: Two sets S1 and S2 are considered equal, given

S1 contains every element from S2.

S2 contains every element from S1.

Theorem: Consider two sets, S1 and S2.

For S1 = S2, it is enough for the following conditions to be satisfied.

for every object x, if \(x \in S1\), then \(x \in S2\).

for every object y, if \(y \in S2\), then \(y \in S1\).

Set Examples:

{1,2,3} - Set of numbers one, two and three.

{5} - A Set containing single element five.

{2+3i, 4-5j} - A Set containing two complex numbers.

__Singleton Set:__

A set containing only one element is called a Singleton set.

Example of singleton set:

{1} -> A Singleton set with 1 member, the number one.

Theorem: {s1} = {s2} only if s1 = s2

\(x \in {s1}\) only if x = s1

__Unordered Pair:__

A set containing two elements is called an unordered pair.

Example: {2,3}

Theorem: {x,y} = {y,x}

\(i \in {x,y}\) only if i = x or i = y.

{x,y} = {a,b} if x=a and y=b OR

x=b and y=a

__Subset:__

Consider two sets, S1 and S2.

Definition of subset: \(S2 \subset S1\) if S1 contains every member contained in S2.

Theorem:

Consider two sets S1 and S2. S1 = S2 only if \(S1 \subset S2\) and \(S2 \subset S1\).