P6 Mathematics - Term I

Topic 1: Sets

Lesson 2: Universal Sets, Complement Sets, and Subsets

Duration: 45 minutes

Today's Learning Objectives for This Lesson

By the end of this lesson, you will be able to:

• Define and identify universal sets in given contexts
• Define and find complement sets
• Understand the concept of subsets
• Form subsets from a given set
• Explain the relationship between universal sets, complement sets, and subsets

Quick Review of Set Types

From Lesson 1, we learned:

• Equal sets (=): Same items, same number
• Equivalent sets (~): Different items, same number
• Unequal sets (≠): Different number of items

Today: We learn about universal sets, complements, and subsets

What is a Universal Set?

The universal set (U) contains ALL items we are considering in a situation.

Examples:

Context: All learners in this classroom
U = {all learners in P6 classroom}

Context: All fruits in a basket
U = {oranges, mangoes, bananas, pineapples}

Symbol: We use U

The Universal Set Depends on Context

Example 1:

U = {all days of the week}
U = {Monday, Tuesday, Wednesday, Thursday, Friday,
Saturday, Sunday}

Example 2:

U = {numbers 1 to 10}
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Key point: The universal set depends on context!

Complement Sets Contain What's Left

Definition: The complement of a set contains all elements in the universal set that are NOT in that set.

Symbol: A' (read as "A prime" or "complement of A")

Example:

U = {1, 2, 3, 4, 5, 6}
A = {2, 4, 6}
A' = {1, 3, 5}

A Picture of a Set and Its Complement

Key relationship: A and A' together equal the universal set U

Practice: Find the Complement of A

U = {a, b, c, d, e, f, g, h}
A = {a, c, e, g}

Question: What is A' (complement of A)?

Think: Which elements are in U but NOT in A?

Answer: The Complement of A

A' = {b, d, f, h}

Check:

• A = {a, c, e, g} (4 elements)
• A' = {b, d, f, h} (4 elements)
• A and A' together = all 8 elements in U ✓

What Makes One Set a Subset?

Definition: A set whose members are ALL contained in another set.

Symbol: ⊂ or ⊆

Example:

U = {all learners in this class}
B = {all boys in this class}

Question: Are all boys also learners? Yes!

Therefore: B ⊂ U (B is a subset of U)

Every Element Must Belong to Both

Key point: Every element in a subset MUST be in the larger set

Many Different Subsets Are Possible

U = {1, 2, ..., 10}

All subsets of U:

• A = {2, 4, 6, 8, 10} ⊂ U
• B = {1, 3, 5, 7, 9} ⊂ U
• C = {1, 2, 3} ⊂ U
• D = {5} ⊂ U

The Empty Set Is Always a Subset

Special case:

E = { } ⊂ U (empty set is a subset of any set)

Why this works:
The empty set has no elements, so there's nothing in it that could fail to
be in U.

Practice: Forming Your Own Subsets

Given: U = {a, b, c, d, e}

Form 3 different subsets from U:

Write them in your exercise book.

Examples could include:

• {a, b}
• {c, d, e}
• {a}

Three Facts Worth Remembering About Subsets

Important facts:

• A set is a subset of itself: U ⊂ U
• The empty set { } is a subset of every set
• If A ⊂ U, then every element of A is also in U

How the Three Ideas Connect

Relationships:

• U contains everything
• A ⊂ U, and A' is the complement of A
• A and A' together equal U

Practice Problem: Apply All Three Ideas

Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
       A = {2, 4, 6, 8, 10}

Answer these:

1. Is A a subset of U?
2. Find A' (complement of A)
3. Is A' also a subset of U?

Checking Your Answers to the Practice Problem

1. Yes, A ⊂ U (all elements of A are in U)
2. A' = {1, 3, 5, 7, 9} (odd numbers)
3. Yes, A' ⊂ U (complement is always a subset of U)

Check: A and A' together = {1, 2, ..., 10} = U ✓

A Real-Life Example Using Your Family

Context: Your family

U = {Father, Mother, Sister1, Sister2, Brother}
A = {female family members} = {Mother, Sister1, Sister2}
A' = {male family members} = {Father, Brother}

Confirming the Family Set Relationships

Notice:

• A ⊂ U (females are family members)
• A' ⊂ U (males are family members)
• A and A' together = U (all family members)

Summary of Today's Three Ideas

Tonight's Homework Assignment on Sets

1. At home, identify a universal set (e.g., all family members, all plates)
2. Create set A (a subset of U)
3. Find A' (complement of A)
4. Given U = {Monday...Sunday}, form 4 different subsets

Expected time: 20 minutes

Preview: Venn Diagrams Next Lesson

Tomorrow we will learn about:

• Venn diagrams
• How to draw sets visually
• Union and intersection of sets

Venn diagrams will help us SEE the relationships we learned today!

Lesson Credits and Curriculum Source

Created: November 21, 2025
Based on: NCDC P6 Mathematics Curriculum - Topic 1: Sets

Source: National Curriculum Development Centre (NCDC), Uganda

Available from: https://ncdc.go.ug/