Explore the basic tenets of sets (types of sets, subsets, operation with sets, and Venn diagrams of two sets)

(a) Listing (Roster) Form

Write all elements inside brackets.

Example: A = {2, 4, 6, 8}

(b) Statement Form

Describe the set in words.

Example: A = {even numbers less than 10}

(c) Set Builder Notation

Write a rule that defines the elements.

Example: A = {x : x is an even number, x < 10}

This reads as "A is the set of all x such that x is an even number and x is less than 10."

Finite Set

A set with a countable number of elements. Example: A = {1, 2, 3, 4, 5}

Infinite Set

A set with an uncountable number of elements. Example: A = {2, 4, 6, 8, ...}

Empty Set (Null Set)

A set with no elements. Denoted by { } or ∅. Example: A = {prime numbers between 31 and 37} = ∅

Singleton Set

A set with exactly one element. Example: A = {5}

Equivalent Sets

Two sets are equivalent if they have the same number of elements (n(A) = n(B)). Example: A = {a, b, c} and B = {1, 2, 3} are equivalent because n(A) = n(B) = 3

Equal Sets

Two sets are equal if they contain exactly the same elements. Example: A = {1, 2, 3} and B = {3, 1, 2} are equal (order doesn't matter)

Subsets

Set A is a subset of set B if every element of A is also in B. Written as A ⊂ B or A ⊆ B.

• Proper subset (⊂): A is contained in B but A ≠ B
• Improper subset (⊆): A is contained in B or A = B
• The empty set ∅ is a subset of every set.
• Every set is a subset of itself.

Example: If A = {1, 2} and B = {1, 2, 3}, then A ⊂ B

Universal Set

The universal set (μ) contains all elements under consideration in a particular problem.

Union (∪)

The union of sets A and B contains all elements that are in A or B or both.

Notation: A ∪ B = {x : x ∈ A or x ∈ B}

Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}

Intersection (∩)

The intersection of sets A and B contains only elements common to both sets.

Notation: A ∩ B = {x : x ∈ A and x ∈ B}

Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}

When sets have no common elements, they are disjoint sets.

Complement (A' or Aᶜ)

The complement of set A contains all elements in the universal set that are not in A.

Notation: A' = {x : x ∈ μ and x ∉ A}

Example: If μ = {1, 2, 3, 4, 5} and A = {1, 2}, then A' = {3, 4, 5}

For any two sets A and B:

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

Also:

$n(\mu) = n(A) + n(A')$

A Venn diagram is a pictorial representation of sets. The universal set is represented by a rectangle, and subsets are represented by circles or ovals.

Drawing a Venn Diagram for Two Sets

1. Draw a rectangle representing the universal set μ
2. Draw two overlapping circles inside the rectangle
3. Label each circle with the set name
4. Place common elements in the overlapping region
5. Place elements unique to each set in their respective circles
6. Place elements outside both sets outside the circles but inside μ

Worked Example

In a class of 30 students, 18 study Mathematics (M), 15 study Physics (P), and 8 study both subjects. Find how many study:

(a) Mathematics only (b) Physics only (c) At least one subject (d) Neither subject

Solution:

• n(M) = 18, n(P) = 15, n(M ∩ P) = 8, n(μ) = 30

(a) Mathematics only = n(M) - n(M ∩ P) = 18 - 8 = 10

(b) Physics only = n(P) - n(M ∩ P) = 15 - 8 = 7

(c) At least one subject = n(M ∪ P) = n(M) + n(P) - n(M ∩ P) = 18 + 15 - 8 = 25

(d) Neither subject = n(μ) - n(M ∪ P) = 30 - 25 = 5

Another Worked Example

A survey of 50 shoppers in Dar es Salaam found that 30 buy rice (R), 25 buy beans (B), and 10 buy neither. How many buy both rice and beans?

Solution:

• n(μ) = 50, n(R) = 30, n(B) = 25, n(R ∪ B)' = 10
• n(R ∪ B) = n(μ) - n(R ∪ B)' = 50 - 10 = 40
• Using: n(R ∪ B) = n(R) + n(B) - n(R ∩ B)
• 40 = 30 + 25 - n(R ∩ B)
• n(R ∩ B) = 55 - 40 = 15

Therefore, 15 shoppers buy both rice and beans.

In Tanzania, market vendors often use set concepts when organizing their goods. For example, a vendor at Mwalimu Nyerere Market in Dar es Salaam might categorize fruits into different sets: set A = {oranges, mangoes, bananas} and set B = {mangoes, pawpaw, apples}. The vendor can use union to know all fruits available, intersection to identify fruits that appear in both categories (for promotional deals), and complement to find which fruits are not in a particular category. This helps in inventory management and planning which fruits to restock.