The union of two sets A and B is the set containing all elements that belong to A, or B, or both. The keyword for union is "or".

Notation: A ∪ B

Definition: x ∈ (A ∪ B) if and only if x ∈ A or x ∈ B or x ∈ both A and B.

Worked Example 1

If A = {a, b, c, d, e, f} and B = {a, e, i, o, u}, find A ∪ B.

Solution

A ∪ B = {a, b, c, d, e, f, i, o, u}

The elements a and e appear in both sets but are written only once in the union.

The intersection of two sets A and B is the set containing only the elements that are common to both A and B. The keyword for intersection is "and".

Notation: A ∩ B

Definition: x ∈ (A ∩ B) if and only if x ∈ A and x ∈ B.

Worked Example 2

If A = {1, 2, 3, 4, 5} and B = {1, 3, 5}, find A ∩ B.

Solution

A ∩ B = {1, 3, 5}

These are the elements that appear in both sets.

Disjoint Sets

Two sets are disjoint if they have no elements in common. In this case, A ∩ B = ∅ (empty set).

Worked Example 3

If A = {a, e, i} and B = {b, c, f}, are A and B disjoint?

Solution

A and B have no common elements, so they are disjoint sets. A ∩ B = ∅

The complement of a set A (denoted A' or Aᶜ) is the set of all elements in the universal set μ that are not in A.

Notation: A' or Aᶜ

Definition: x ∈ A' if and only if x ∈ μ and x ∉ A.

Worked Example 4

Given μ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 3, 5, 7}, find A'.

Solution

A' = {1, 4, 6, 8, 9, 10}

These are all elements in the universal set that are not in A.

Worked Example 5

Given μ = {a, c, i, o, u} and B = {c, i}, find B'.

Solution

B' = {a, o, u}

These are elements in μ that are not in B.

Given μ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, A = {2, 3, 5, 7, 11, 13}, and B = {2, 4, 6, 8, 10, 12, 14}, find:

(a) A' ∪ B' (b) A ∩ B'

Solution

First, find the complements:

• A' = {1, 4, 6, 8, 9, 10, 12, 14}
• B' = {1, 3, 5, 7, 9, 11, 13}

(a) A' ∪ B' = {1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}

(b) A ∩ B' = {3, 5, 7, 11, 13}

For any two sets A and B:

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$

Also, for a universal set μ: $n(\mu) = n(A) + n(A')$

In everyday life in Tanzania, set operations are useful when organizing groups. For example, a shopkeeper in a local market in Dar es Salaam might use union to find all customers who bought either rice or beans, use intersection to find customers who bought both rice and beans, and use complement to find customers who did not buy beans at all. This helps in inventory planning and understanding customer purchasing patterns.