Sets

Lessons

Senior 1 · Mathematics

Sets

5 lessons

Lesson 2 of 5

Subsets, the universal set and the complement

24 min

A set inside a set

In one S1 class at a school in Huye, the learners who sing in the school choir are

C = \{\text{Aline},\ \text{Eric},\ \text{Grace}\},

and the learners who go to the Friday music club are

F = \{\text{Aline},\ \text{Eric},\ \text{Grace},\ \text{Kevin},\ \text{Olivier}\}.

Every choir singer is also in the music club, so the whole of $C$ sits inside $F$ . This relationship between two sets has a name and a symbol.

Definition

Subset: $A$ is a subset of $B$ if every element of $A$ is also an element of $B$ .

Notation: $A \subseteq B$ . If also $A \neq B$ , so that $B$ has at least one element that $A$ lacks, $A$ is a proper subset of $B$ , written $A \subset B$ . "Is not a subset of" is written $A \not\subseteq B$ . Many books, including Rwanda's S1 textbook, also write $\subset$ for any subset; the context tells you which is meant.

Example: $C \subset F$ above. Also $\{2, 4\} \subset \{1, 2, 3, 4\}$ , but $\{1, 5\} \not\subseteq \{1, 2, 3, 4\}$ , because $5 \notin \{1, 2, 3, 4\}$ .

Important

Two subsets that every set has

Every set is a subset of itself: $A \subseteq A$ . Every element of $A$ is, of course, in $A$ .
The empty set is a subset of every set: $\varnothing \subseteq A$ .
Why: to show that $\varnothing \not\subseteq A$ , you would have to find an element of $\varnothing$ that is not in $A$ . The empty set has no elements at all, so nothing can break the rule.

Warning

$\in$ or $\subseteq$ ? The symbol $\in$ links an element to a set. The symbol $\subseteq$ links a set to a set.

Statement	True or false?	Reason
$3 \in \{1, 3, 5\}$	True	3 is one of the elements
$\{3\} \subseteq \{1, 3, 5\}$	True	every element of ${3}$ is in $\{1, 3, 5\}$
$3 \subseteq \{1, 3, 5\}$	False	3 is a number, not a set
$\{3\} \in \{1, 3, 5\}$	False	the elements are numbers; the set ${3}$ is not one of them

How many subsets does a set have?

List the subsets of small sets and look for a pattern.

Set	Its subsets	Number of subsets
$\varnothing$	$\varnothing$	1
${a}$	$\varnothing,\ \{a\}$	2
${a, b}$	$\varnothing,\ \{a\},\ \{b\},\ \{a, b\}$	4
$\{a, b, c\}$	$\varnothing,\ \{a\},\ \{b\},\ \{c\},\ \{a, b\},\ \{a, c\},\ \{b, c\},\ \{a, b, c\}$	8

Each new element doubles the number of subsets. The tree below shows why.

To build a subset of $\{a, b, c\}$ , decide for each element whether it is in or out. Three decisions with 2 choices each give $2 \times 2 \times 2 = 8$ subsets.

Important

Rule: a set with $n$ elements has

2^n \text{ subsets} \quad\text{and}\quad 2^n - 1 \text{ proper subsets.}

Why: building a subset means deciding, element by element, "in" or "out". There are 2 choices for each of the $n$ elements, and every combination of choices gives a different subset, so the number of subsets is $2 \times 2 \times \cdots \times 2 = 2^n$ . Exactly one of those subsets is the set itself, so there are $2^{n} - 1$ proper subsets.

Worked example 1: listing subsets

The flag of Rwanda has three colours, $R = \{\text{blue},\ \text{yellow},\ \text{green}\}$ . List every subset of $R$ and count the proper subsets.

Solution Work by size, so that none is missed:

0 colours: $\varnothing$
1 colour: $\{\text{blue}\},\ \{\text{yellow}\},\ \{\text{green}\}$
2 colours: $\{\text{blue}, \text{yellow}\},\ \{\text{blue}, \text{green}\},\ \{\text{yellow}, \text{green}\}$
3 colours: $\{\text{blue}, \text{yellow}, \text{green}\}$

That is $1 + 3 + 3 + 1 = 8 = 2^{3}$ subsets, as the rule predicts. All except $R$ itself are proper, so there are $2^{3} - 1 = 7$ proper subsets.

Worked example 2: using the rule backwards

(a) A set $S$ has 64 subsets. How many elements does it have?
(b) How many subsets has $P = \{x : x \text{ is a prime number less than } 12\}$ ?

Solution

(a) $2^{n} = 64 = 2^{6}$ , so $n = 6$ . $S$ has 6 elements (and $64 - 1 = 63$ proper subsets).

(b) $P = \{2, 3, 5, 7, 11\}$ , so $n (P) = 5$ and $P$ has $2^{5} = 32$ subsets.

The universal set and Venn diagrams

Definition

Universal set: the set of all the elements under discussion in a problem. Every other set in the problem is a subset of it.

Notation: $U$ . Some books use ξ or ε.

Example: in a question about the learners of S1 B, $U$ is the set of all learners in S1 B. The girls, the choir members and the footballers of S1 B are all subsets of $U$ .

Definition

Venn diagram: a picture of sets in which the universal set is drawn as a rectangle and each set as a closed curve (usually a circle) inside it. Each element is written inside every curve of the sets it belongs to.

Example: the figure below, with $U = \{1, 2, 3, \ldots, 12\}$ .

The diagrams are named after John Venn (1834–1923), an English logician who made them popular in a paper of 1880.

$U = \{1, 2, \ldots, 12\}$ , $A = \{1, 2, 3, 4, 6, 12\}$ (the factors of 12) and $B = \{1, 2, 4\}$ (the factors of 4). Every factor of 4 is also a factor of 12, so $B \subset A$ and circle $B$ is drawn inside circle $A$ .

The complement of a set

Definition

Complement of a set $A$ : the set of all the elements of $U$ that are not in $A$ .

Notation: $A^{'}$ (also written $A^{c}$ ):

A' = \{x : x \in U \text{ and } x \notin A\}.

Example: if $U = \{a, b, c, d, e\}$ and $A = \{a, e\}$ , then $A' = \{b, c, d\}$ .

The shaded region is $A^{'}$ : everything in $U$ outside $A$ .

Important

Rule: $n (A^{'}) = n (U) - n (A)$ .

Why: every element of $U$ is in exactly one of $A$ and $A^{'}$ , never in both and never in neither. So the two counts add up to the whole: $n (A) + n (A^{'}) = n (U)$ .

Worked example 3: finding a complement

$U = \{x : x \text{ is a whole number},\ 1 \le x \le 12\}$ and $A$ is the set of multiples of 3 in $U$ . Find $A^{'}$ and check the rule $n (A^{'}) = n (U) - n (A)$ .

Solution $A = \{3, 6, 9, 12\}$ , so

A' = \{1, 2, 4, 5, 7, 8, 10, 11\}.

Check: $n (A^{'}) = 8$ and $n (U) - n (A) = 12 - 4 = 8$ . ✓

Tip

Three facts follow straight from the definition: $(A^{'})^{'} = A$ , $U' = \varnothing$ and $\varnothing' = U$ . Taking the complement twice brings you back where you started, like turning round twice.

Khan Academy: Subset, strict subset, and superset (5 min). Watch how Sal tells a subset from a strict (proper) subset, and listen for the word superset: B is a superset of A when A is a subset of B. His symbol for a strict subset may differ from the ⊂ used in this lesson; the idea is the same.

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Practice

Try each one before you open the answers.

Write $\in$ , $\notin$ , $\subseteq$ or $\not\subseteq$ in each gap: (a) $5 \ \_\_\ \{1, 3, 5\}$ (b) $\{1, 5\} \ \_\_\ \{1, 3, 5\}$ (c) $\{2\} \ \_\_\ \{1, 3, 5\}$ (d) $\varnothing \ \_\_\ \{1, 3, 5\}$
List all the subsets of ${x, y}$ . Which of them are proper subsets?
How many subsets, and how many proper subsets, does the set of the letters of the word KIVU have?
A set has 128 subsets. How many elements does it have?
$U = \{1, 2, \ldots, 15\}$ and $P$ is the set of prime numbers in $U$ . Find $P^{'}$ and check that $n (P^{'}) = n (U) - n (P)$ .
$U = \{a, b, c, d, e, f\}$ and $A = \{a, c, e\}$ . Find $A^{'}$ , $(A^{'})^{'}$ and $U^{'}$ .
How many subsets of $\{1, 2, 3, 4, 5\}$ contain the element 1? Explain your reasoning.

Show answersHide answers

(a) $\in$ (b) $\subseteq$ (c) $\not\subseteq$ , since $2 \notin \{1, 3, 5\}$ (d) $\subseteq$
$\varnothing,\ \{x\},\ \{y\},\ \{x, y\}$ . All except ${x, y}$ are proper.
$\{K, I, V, U\}$ has 4 elements: $2^{4} = 16$ subsets and 15 proper subsets.
$2^{n} = 128 = 2^{7}$ , so 7 elements.
$P = \{2, 3, 5, 7, 11, 13\}$ , so $P' = \{1, 4, 6, 8, 9, 10, 12, 14, 15\}$ . Check: $9 = 15 - 6$ . ✓
$A' = \{b, d, f\}$ , $(A')' = \{a, c, e\} = A$ and $U' = \varnothing$ .
Put 1 in, then decide freely for each of the other 4 elements: $2^{4} = 16$ subsets.

Sources and credits

1.2: Subsets, 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: Subset, strict subset, and superset, Khan Academy, embedded from YouTube.

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

1A set inside a set

2How many subsets does a set have?

3The universal set and Venn diagrams

4The complement of a set

A set inside a set

How many subsets does a set have?

The universal set and Venn diagrams

The complement of a set