Basics of Set Theory

 Set theory basics:

Example set: 

Set of numbers less than 5 but greater than 1 = {2, 3, 4, 5}. Call the set "S".

Notation used:

{2, 3, 4, 5}

Read as: The set that contains the numbers 2, 3, 4, 5.

2 ∈ S and 6 ∉ S

Read as: 2 is an element of set S and 6 is not an element of S. 

Subset: B is a subset of A if all elements of B are elements of A.

Example subset:

Notation used:

B ⊆ A

Read as: B is a subset of A.
Proper subset: B is a proper subset of A if all elements of B are elements of A and A has elements that B does not.

Example of proper subsets:

A = {1, 2, 3, 4, 5, 6}, B = {1, 2, 3}. B is a proper subset of A since every member of B is a member of A and A has members that B does not.

Notation used:

B ⊂ A

Read as: B is a proper subset of A.

Power set:  The power set of any set A is the set of all possible subsets of A.

Example of power set:

S = {1, 2, 3}

Power set of S = {{∅}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}}

Notation used:

ℙ(S) 

Set Union: The union of two or more sets is the set that contains all and only members of the sets in question.

Example of set union:

A = {1, 2, 3}, B = {4, 5, 6}

The union of A and B = {1, 2, 3, 4, 5, 6}

Notation used:

A∪B = The union of A and B

Set intersection: The set of all common elements between two or more sets.

Example of set intersection:

A = {1, 2, 3}, B = {3, 4, 5} 

The intersection of A and B = {3}

Notation used:

A∩B = The intersection of A and B.