Set Operations


∪ - Union: Given sets A and B, the union of the sets A and B, denoted by A ∪ B, is the set containing the elements in A or B or both.

A ∪ B = {x | x ∈ A ∨ x ∈ B}

Ex.
{1, 3, 5} ∪ {1, 2, 3} = {1, 2, 3, 5}


∩ - Intersection: Given sets A and B, the intersection A and B, denoted by A ∩ B, is the set contaioning the elements in both A and B.

A ∩ B = {x | x ∈ A ∧ x ∈ B}

Ex.
{1, 3, 5} ∩ {1, 2, 3} = {1, 3}


Disjoint - two sets are called disjoint if their intersection is the empty set.

Ex.
{1, 3} ∩ {2, 4} = ∅


Difference - Given sets A and B, the difference of A and B, denoted by A - B, is the set containing the elements in A but not in B. Also can be called the complement of B with respect to A.

A - B = {x | x ∈ A ∧ x ∉ B}

Ex:
{1, 3, 5} - {1, 2, 3} = {5}


Complement - Given U is the universal set, the complement of the set A, denoted by A', is the difference of U and A. Can also be referred to as the complement of A with respect to U. The complement of the set A is U - A.
Note, A' often is written as A with a line above it. The reason A' is being used, is due to lack of a font where I can display a bar above the A.

A' = {x | x ∉ A}

Ex.
Let A = {a, e, i, o, u} and the universal set U be the letters of the English alphabet.
A' = {b, c, d, f, g ... , z}
(A' would contain the all the alphabet except for the vowels)

Ex.
Let A be the set of positive integers greater than 10 and the universal set U be the set of all positive integers. A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}