Sets III

Grade 10 Math - Sets and Venn Diagrams

Lesson Objectives

Understand what a set is and represent sets using different notations.
Identify different types of sets (finite, infinite, null, universal, etc.).
Use set builder notation correctly.
Solve problems involving union, intersection, and complement of sets.
Interpret and draw Venn diagrams involving two or three sets.

Lesson Introduction

Sets are collections of distinct objects. In mathematics, we use sets to group elements and perform logical operations like union, intersection, and complement. Venn diagrams are visual tools for representing these operations clearly.

Core Lesson Content

1. Set Notation

Sets are usually represented by capital letters, and their elements are written within curly brackets.

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

2. Types of Sets

Finite Set: Has a countable number of elements.
Infinite Set: Has uncountable elements (e.g. set of all natural numbers).
Null Set: A set with no elements, denoted $\emptyset$ or $\{\}$ .
Universal Set: The set containing all elements under consideration.
Subset: A set whose elements all belong to another set.

3. Set Builder Notation

Used to describe the elements of a set using a rule.

Example: $B = \{ x \mid x \text{ is an even number less than } 10 \}$

4. Set Operations

Union ( $\cup$ ): All elements in either set A or B or both.
Intersection ( $\cap$ ): Elements common to both sets A and B.
Complement ( $A'$ or $A^c$ ): Elements in the universal set but not in A.
Difference ( $A - B$ ): Elements in A but not in B.

5. Venn Diagrams

Venn diagrams use overlapping circles to show relationships between sets.

They help visualize operations and solve word problems, especially with two or three sets.

Worked Example

Example 1: If

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

, find

A \cup B

A \cup B = \{1, 2, 3, 4, 5\}

Example 2: Find

A \cap B

from the sets above.

A \cap B = \{3\}

Example 3: Given

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

, find

A'

A' = \{1,3,5\}

Example 4: Represent as set

A = \{x \mid x \lt 5,\ x \in \mathbb{N} \}

A = \{1, 2, 3, 4\}

Example 5: In a survey, 30 students like Maths, 20 like English, and 10 like both. How many like either Maths or English?

n(M \cup E) = 30 + 20 - 10 = 40

Example 6: Use a Venn diagram to solve:
In a group of 50 students, 25 play football, 20 play basketball, and 10 play both. How many play neither?

n(F \cup B) = 25 + 20 - 10 = 35

; So,

50 - 35 = 15

students play neither.

Example 7: If

A = \{2, 4, 6, 8\}, B = \{4, 8, 10\}

, find

A - B

A - B = \{2, 6\}

Example 8: [WAEC] In a class, 35 students take Maths, 30 take Biology, and 20 take both. How many take only Maths?

35 - 20 = 15

Example 9: Draw a Venn diagram for three sets A, B, and C where:

n(A) = 40, n(B) = 30, n(C) = 20, n(A \cap B) = 10, n(B \cap C) = 5, n(A \cap C) = 8, n(A \cap B \cap C) = 3

.
Use inclusion-exclusion to solve.

Example 10: In a group of 100 students:
60 take Physics, 50 take Chemistry, 40 take Biology,
25 take both Physics and Chemistry, 20 take both Chemistry and Biology,
15 take both Physics and Biology, and 10 take all three subjects.
How many students take at least one subject?

n(P \cup C \cup B) = 60 + 50 + 40 - 25 - 20 - 15 + 10 = 100

Exercises

List all subsets of the set $\{1, 2\}$ .
[WAEC] Find $A \cup B$ and $A \cap B$ if $A = \{a, b\}, B = \{b, c\}$ . (Past Question)
If $U = \{1,2,3,4,5,6,7,8\}, A = \{2,4,6,8\}$ , find $A'$ .
Write the set of all odd numbers less than 10 using set builder notation.
[NECO] In a class of 60 students, 35 offer French, 30 offer Spanish, and 10 offer both. How many students offer neither? (Past Question)
Use a Venn diagram to solve: $n(A) = 15, n(B) = 20, n(A \cap B) = 5$ . Find $n(A \cup B)$ .
[JAMB] Find $A - B$ for $A = \{1,3,5,7\}, B = \{3,5\}$ . (Past Question)
Use a three-set Venn diagram to represent: 70 take Economics, 60 take Government, 50 take Literature, 20 take all three, 15 each take two subjects only. Find those who take at least one subject.
[WAEC] Represent the set of prime numbers less than 20 using roster notation. (Past Question)
Draw and label a Venn diagram showing: $A \cup B = U$ , $A \cap B = \emptyset$ .

Conclusion/Recap

In this lesson, we learned about sets and set notation, explored key operations like union, intersection, and complement, and practiced solving real-world problems using Venn diagrams. Understanding these fundamentals is critical for further studies in logic, probability, and statistics.