Set Operations and Complements Revision Notes for HSC SSCE Mathematics Advanced

Set Operations and Complements

Introduction

This note covers fundamental set operations including intersection and union, as well as the concept of disjoint sets. Understanding these operations is essential for solving probability problems and working with sets in mathematics.

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

Find the intersection or union of two or more sets
Define disjoint sets as having an empty intersection
Identify whether given sets are disjoint
Apply multiple set operations to solve problems

Intersection and union of sets

Set operations allow us to combine sets in different ways. The two most important operations are intersection and union.

Intersection

The intersection of two sets finds the elements that appear in both sets.

For sets $A$ and $B$ , the intersection is the set of elements that are in both $A$ and in $B$ . The intersection of $A$ and $B$ is denoted $A \cap B$ .

Example: If $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$ , then:

$A \cap B = \{2, 3\}$

This result contains only the elements that appear in both sets.

Union

The union of two sets combines all elements from both sets, without repetition.

For sets $A$ and $B$ , the union is the set of elements which are in $A$ or $B$ or both, and is written as $A \cup B$ , read as " $A$ union $B$ ".

Example: If $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$ , then:

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

This result contains all unique elements from both sets, with duplicates removed.

infoNote

Understanding Intersection vs Union:

Think of intersection as "AND" - an element must be in both sets. Think of union as "OR" - an element can be in either set or both sets. This distinction helps you remember which operation to use for different problems.

Visual representation

Venn diagrams help visualise set operations:

For union ( $A \cup B$ ), we shade both circles completely, representing all elements in either set
For intersection ( $A \cap B$ ), we shade only the overlapping region where the circles overlap, representing elements in both sets

lightbulbExample

Worked Example: Finding Intersection and Union

Question: Let the universal set be $U = \{1, 2, 3, 4, 5, 6\}$ , with sets $A = \{1, 3, 5\}$ and $B = \{2, 3, 4\}$ . Find:

a) $A \cap B$

b) $A \cup B$

c) $\overline{A}$

Solution:

Part a: $A \cap B$

Strategy: Identify elements common to both $A$ and $B$ .

Working:

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

List elements in both sets.

$= \{3\}$

The only common element is 3.

Part b: $A \cup B$

Strategy: Combine all elements from $A$ and $B$ , removing duplicates.

Working:

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

List elements from both sets.

$= \{1, 2, 3, 4, 5\}$

Combine all unique elements, removing the duplicate 3.

Part c: $\overline{A}$

Strategy: List elements in $U$ that are not in $A$ .

Working:

$\overline{A} = \{1, 2, 3, 4, 5, 6\} \setminus \{1, 3, 5\}$

Identify elements in $U$ not in $A$ .

$= \{2, 4, 6\}$

List the remaining elements.

Check: Verify that all unique elements from both sets are included without repetition in the union, and that only common elements appear in the intersection.

Key idea: The intersection $A \cap B$ contains elements in both $A$ and $B$ . The union $A \cup B$ contains elements in $A$ , $B$ , or both.

Disjoint sets

Sometimes two sets have no elements in common. These sets are called disjoint sets.

Definition

Disjoint sets are two sets which do not have any common elements.

Two sets $A$ and $B$ are disjoint if they have no elements in common, meaning their intersection is the empty set.

chatImportant

Mathematical Condition for Disjoint Sets:

$A \cap B = \varnothing$

This is the key condition for sets $A$ and $B$ to be disjoint. If their intersection equals the empty set, the sets are disjoint.

Example: If $A = \{1, 3, 5\}$ and $B = \{2, 4, 6\}$ , then $A \cap B = \varnothing$ , so $A$ and $B$ are disjoint.

Conversely, if $A$ and $B$ are not disjoint, they share at least one element, so $A \cap B \neq \varnothing$ .

lightbulbExample

Worked Example: Determining Disjoint Sets

Question: Let the universal set be $U = \{a, b, c, d, e\}$ , with sets $A = \{a, b\}$ and $B = \{c, d\}$ . Determine:

a) If $A$ and $B$ are disjoint

b) $A \cup B$

Solution:

Part a: If $A$ and $B$ are disjoint

Strategy: Check if $A \cap B = \varnothing$ .

Working:

$A \cap B = \{a, b\} \cap \{c, d\}$

List elements in both sets.

$= \varnothing$

There are no common elements.

Since $A \cap B = \varnothing$ , A and B are disjoint.

Part b: $A \cup B$

Strategy: Combine elements from $A$ and $B$ , ensuring no duplicates.

Working:

$A \cup B = \{a, b\} \cup \{c, d\}$

List elements from both sets.

$= \{a, b, c, d\}$

Combine all elements without duplicates.

Key idea: Sets $A$ and $B$ are disjoint if $A \cap B = \varnothing$ , meaning they have no elements in common. The union $A \cup B$ of disjoint sets includes all elements from both sets.

Exam tips

chatImportant

Common Mistakes to Avoid:

Always check carefully which elements belong to each set before finding intersections or unions
When finding unions, remember to list each element only once, even if it appears in both sets
To test if sets are disjoint, find their intersection - if it's empty, they're disjoint
Venn diagrams can be helpful for visualising set operations, especially with more complex problems

bookmarkSummary

Key Points to Remember:

The intersection $A \cap B$ contains only elements that are in both sets $A$ and $B$
The union $A \cup B$ contains all elements that are in set A, set B, or both, listed without repetition
Sets are disjoint when they have no common elements, expressed mathematically as $A \cap B = \varnothing$
When combining sets using union, always ensure each element is listed only once
The intersection is about what sets share in common (AND), while the union is about combining what either set contains (OR)

Set Operations and Complements (HSC SSCE Mathematics Advanced): Revision Notes