Sets

Lessons

Senior 1 · Mathematics

Sets

5 lessons

Lesson 3 of 5

Intersection, union and difference of sets

25 min

Two clubs, three questions

In an S1 class at a school in Rubavu, the debate club and the science club have these members:

D = \{\text{Aline},\ \text{Brian},\ \text{Chantal},\ \text{Didier},\ \text{Esther}\}

S = \{\text{Chantal},\ \text{Esther},\ \text{Fabrice},\ \text{Gloria}\}

The class teacher wants to know: Who is in both clubs? Who is in at least one club? Who is in the debate club only? Each question asks for a new set made from $D$ and $S$ . The three ways of making it are called intersection, union and difference.

Definition

Intersection of $A$ and $B$ : the set of elements that are in $A$ and in $B$ .

A \cap B = \{x : x \in A \text{ and } x \in B\}

Read $A \cap B$ as "A intersection B" or "A and B".

Example: $D \cap S = \{\text{Chantal},\ \text{Esther}\}$ .

$A \cap B$ : only the overlap is shaded.

Definition

Disjoint sets: two sets with no element in common, so that $A \cap B = \varnothing$ . In a Venn diagram their circles are drawn apart.

Example: in $U = \{1, 2, \ldots, 8\}$ , the odd numbers $\{1, 3, 5, 7\}$ and the even numbers $\{2, 4, 6, 8\}$ are disjoint.

Disjoint sets: $A \cap B = \varnothing$ .

Definition

Union of $A$ and $B$ : the set of elements that are in $A$ or in $B$ (or in both).

A \cup B = \{x : x \in A \text{ or } x \in B\}

Read $A \cup B$ as "A union B".

Example: $D \cup S = \{\text{Aline},\ \text{Brian},\ \text{Chantal},\ \text{Didier},\ \text{Esther},\ \text{Fabrice},\ \text{Gloria}\}$ . Chantal and Esther are in both clubs, but each name is written once.

$A \cup B$ : everything inside either circle is shaded.

Definition

Difference of $A$ and $B$ : the set of elements that are in $A$ but not in $B$ .

A - B = \{x : x \in A \text{ and } x \notin B\}

Notation: $A - B$ , also written $A \setminus B$ . Notice that $A - B = A \cap B'$ .

Example: $D - S = \{\text{Aline},\ \text{Brian},\ \text{Didier}\}$ , the learners in the debate club only, while $S - D = \{\text{Fabrice},\ \text{Gloria}\}$ .

Definition

Symmetric difference of $A$ and $B$ : the set of elements that are in exactly one of the two sets.

A \,\Delta\, B = (A - B) \cup (B - A) = (A \cup B) - (A \cap B)

Example: $D \,\Delta\, S = \{\text{Aline},\ \text{Brian},\ \text{Didier},\ \text{Fabrice},\ \text{Gloria}\}$ .

Every operation as a shaded region

The shaded region in each diagram is the set named under it.

Worked example 1: all the operations on two sets

$U = \{1, 2, 3, \ldots, 12\}$ , $A$ is the set of the factors of 12 and $B$ is the set of the prime numbers in $U$ . Draw a Venn diagram, then find $A \cap B$ , $A \cup B$ , $A - B$ , $B - A$ , $A \,\Delta\, B$ and $(A \cup B)'$ .

Solution First list the sets: $A = \{1, 2, 3, 4, 6, 12\}$ and $B = \{2, 3, 5, 7, 11\}$ . Place the common elements, 2 and 3, in the overlap first; then the rest of each set; then the elements of $U$ that are in neither set.

Factors of 12 (A) and primes up to 12 (B).

$A \cap B = \{2, 3\}$
$A \cup B = \{1, 2, 3, 4, 5, 6, 7, 11, 12\}$
$A - B = \{1, 4, 6, 12\}$ and $B - A = \{5, 7, 11\}$
$A \,\Delta\, B = \{1, 4, 5, 6, 7, 11, 12\}$
$(A \cup B)' = \{8, 9, 10\}$

Check: $n(A \cup B) = 9$ , and $n(A) + n(B) - n(A \cap B) = 6 + 5 - 2 = 9$ . You will use this check again in the next lesson.

Worked example 2: reading a Venn diagram

Read each set from the regions.

From the diagram, list $U$ , $A$ , $B$ , $A^{'}$ , $A \cap B'$ and $(A \cup B)'$ .

Solution A set is everything inside its circle, including the overlap.

$U = \{p, q, r, s, t, u, v, w\}$
$A = \{p, q, r\}$ and $B = \{r, s, t, u\}$
$A' = \{s, t, u, v, w\}$ : everything outside circle $A$
$A \cap B' = \{p, q\}$ : inside $A$ and outside $B$ , which is the same set as $A - B$
$(A \cup B)' = \{v, w\}$

Important

Rule: if $A \subseteq B$ , then $A \cap B = A$ and $A \cup B = B$ .

Why: every element of $A$ is already in $B$ . So the elements in both sets are just the elements of $A$ , and joining $A$ to $B$ adds nothing new. For example, $\{2, 4\} \cap \{1, 2, 3, 4, 5\} = \{2, 4\}$ and $\{2, 4\} \cup \{1, 2, 3, 4, 5\} = \{1, 2, 3, 4, 5\}$ .

Warning

"Or" includes "both". An element in both sets belongs to $A \cup B$ , and it is written only once.
Order matters in a difference. $A - B$ and $B - A$ are usually different sets, as in Worked example 1.
$A \cap B$ is a set, not a number. Write $A \cap B = \{2, 3\}$ , and $n(A \cap B) = 2$ for its number of elements.

Set notation at a glance

The examples use $U = \{1, 2, \ldots, 8\}$ , $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$ .

Symbol	Meaning	Example
$\in$	is an element of	$2 \in A$
$\notin$	is not an element of	$5 \notin A$
$n (A)$	number of elements of $A$	$n (A) = 4$
$\varnothing$	the empty set	$A \cap \{7, 8\} = \varnothing$
$\subseteq$	is a subset of	$\{3, 4\} \subseteq B$
$\subset$	is a proper subset of	$\{1, 2\} \subset A$
$\not\subseteq$	is not a subset of	$B \not\subseteq A$
$U$	the universal set	$U = \{1, 2, \ldots, 8\}$
$A^{'}$	complement of $A$	$A' = \{5, 6, 7, 8\}$
$\cap$	intersection ("and")	$A \cap B = \{3, 4\}$
$\cup$	union ("or")	$A \cup B = \{1, 2, 3, 4, 5, 6\}$
$-$	difference	$A - B = \{1, 2\}$
$\Delta$	symmetric difference	$A \,\Delta\, B = \{1, 2, 5, 6\}$

Tip

The cup-shaped $\cup$ collects everything, like a cup that holds the contents of both sets: U for Union. The $\cap$ is the cup turned upside down, covering only what the two sets share.

Khan Academy: Intersection and union of sets (8 min). Watch how an element that belongs to both sets is written only once in the union, and how the intersection keeps only the shared elements.

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Practice

Questions 1 to 3 use $U = \{1, 2, \ldots, 10\}$ , $A = \{1, 2, 3, 4, 5\}$ and $B = \{2, 4, 6, 8, 10\}$ .

Find (a) $A \cap B$ (b) $A \cup B$ (c) $A - B$ (d) $B^{'}$ (e) $(A \cup B)'$ .
Draw a Venn diagram of $U$ , $A$ and $B$ , placing every element of $U$ .
Show that $(A \cup B)' = A' \cap B'$ for these sets.
$P$ is the set of the letters of KIGALI and $Q$ the set of the letters of GISENYI. Find $P \cap Q$ , $P \cup Q$ and $P \,\Delta\, Q$ .
Are the set of multiples of 4 and the set of odd numbers disjoint? Explain.
$A \subseteq B$ , $n (A) = 5$ and $n (B) = 9$ . Find $n(A \cap B)$ , $n(A \cup B)$ and $n (B - A)$ .
Use the diagram in Worked example 2 to list $A' \cap B$ . Which other set in this lesson is equal to it?

Show answersHide answers

(a) ${2, 4}$ (b) $\{1, 2, 3, 4, 5, 6, 8, 10\}$ (c) $\{1, 3, 5\}$ (d) $\{1, 3, 5, 7, 9\}$ (e) ${7, 9}$
$A$ only: 1, 3, 5; overlap: 2, 4; $B$ only: 6, 8, 10; outside both circles: 7, 9.
$A' = \{6, 7, 8, 9, 10\}$ and $B' = \{1, 3, 5, 7, 9\}$ , so $A' \cap B' = \{7, 9\} = (A \cup B)'$ . ✓
$P = \{K, I, G, A, L\}$ , $Q = \{G, I, S, E, N, Y\}$ : $P \cap Q = \{G, I\}$ , $P \cup Q = \{K, I, G, A, L, S, E, N, Y\}$ , $P \,\Delta\, Q = \{K, A, L, S, E, N, Y\}$ .
Yes. Every multiple of 4 is even, so no number is in both sets: the intersection is $\varnothing$ .
$A \cap B = A$ , so 5; $A \cup B = B$ , so 9; $n (B - A) = 9 - 5 = 4$ .
$A' \cap B = \{s, t, u\}$ , which equals $B - A$ .

Sources and credits

1.4: Set Operations with Two Sets, from Contemporary Mathematics by Donna Kirk and contributors, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
1.3: Understanding Venn Diagrams, from Contemporary Mathematics by Donna Kirk and contributors, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
Video: Intersection and union of sets, Khan Academy, embedded from YouTube.

The Rwandan worked examples, the diagrams, the practice items and the quiz questions were written by Ganzaa.

1Two clubs, three questions

2Every operation as a shaded region

3Set notation at a glance

Two clubs, three questions

Every operation as a shaded region

Set notation at a glance