Sets, Subsets & Power Sets — notes, question types & practice · Set Theory & Venn Diagrams

01

What a set is

A set is a collection of well-defined objects. The objects are called its elements.

Write a set in roster form: $A = \{2, 4, 6, 8\}$ . Then $n(A) = 4$ , the number of elements.

$4 \in A$ means 4 belongs to A.
$5 \notin A$ means 5 does not belong to A.

Two sets are equal when they have exactly the same elements. They are equivalent when they only have the same number of elements. $\{1, 2\}$ and $\{3, 4\}$ are equivalent but not equal.

The empty set $\emptyset$ has no elements, so $n(\emptyset) = 0$ . A singleton set has exactly one element.

02

Subsets and the power set

$A$ is a subset of $B$ (written $A \subseteq B$ ) when every element of A is also in B.

Each element makes two choices: in or out. So a set with $n$ elements has $2^n$ subsets.

The set of all subsets is the power set. A power set of an $n$ -element set has $2^n$ elements.

Subsets: $2^n$
Proper subsets (drop the set itself): $2^n - 1$
Non-empty subsets (drop the empty set): $2^n - 1$
Non-empty proper subsets: $2^n - 2$

Rule: For $n$ elements: subsets $2^n$ , proper $2^n - 1$ , non-empty proper $2^n - 2$ .

Check with $A = \{1, 2, 3\}$ : the 8 subsets are $\emptyset$ , three singletons, three pairs and the full set. Proper: 7. Non-empty proper: 6.

03

Subsets of a fixed size

Subsets with exactly $r$ elements are counted by $^{n}C_r$ .

From $\{1, 2, 3, 4, 5, 6\}$ , the subsets of size 2 number $^{6}C_2 = \dfrac{6 \times 5}{2} = 15$ .

To count subsets that must contain a given element, put that element in first and choose the rest from the remaining $n - 1$ . To count subsets that must avoid an element, choose everything from the other $n - 1$ .

Tip: "At least one of two given elements" flips to "avoid both". Count the avoid case and subtract from $2^n$ .

04

Union, intersection, difference, complement

Take $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5\}$ .

Union $A \cup B = \{1, 2, 3, 4, 5\}$ , so $n(A \cup B) = 5$ : in A or B or both.
Intersection $A \cap B = \{3, 4\}$ , so $n(A \cap B) = 2$ : in both.
Difference $A - B = \{1, 2\}$ : in A but not in B.
Complement $A' = U - A$ : everything of the universal set $U$ outside A.

Rule: $n(A - B) = n(A) - n(A \cap B)$ . Subtract the overlap, not $n(B)$ .

05

The union counting formula

n(A \cup B) = n(A) + n(B) - n(A \cap B)

Adding $n(A)$ and $n(B)$ counts the common elements twice. Subtract the overlap once.

Worked: $n(A) = 23$ , $n(B) = 18$ , $n(A \cap B) = 9$ . Union $= 23 + 18 - 9 = 32$ .

The formula also runs backwards. If the union is known, any one of the other three counts follows from it.

Tip: "At least one of A or B" always means the union. Spot those words and reach for this formula.

06

De Morgan's laws

The complement flips unions and intersections:

(A \cup B)' = A' \cap B' \qquad (A \cap B)' = A' \cup B'

In words: "not in A or B" means "outside A and outside B".

Exam use: $n(A' \cap B') = n(U) - n(A \cup B)$ .

Worked: $n(U) = 50$ , $n(A) = 30$ , $n(B) = 24$ , $n(A \cap B) = 10$ . Union $= 44$ , so $n(A' \cap B') = 50 - 44 = 6$ .

Watch: A complement turns $\cup$ into $\cap$ and $\cap$ into $\cup$ . Do not keep the same join sign.

07

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Number of subsets of a set

How to spot it:

'How many subsets does a set with n elements have?' — a direct power-set ask.

2^n

Method

Count the elements $n$ .
Compute $2^n$ .
For proper subsets subtract 1.

Why it works:

Each element independently joins or stays out, so the choices double for every element.

Try this

A set has 6 elements. How many subsets does it have?

Show solution

Count the elements: $n = 6$ .
Subsets $= 2^6 = 64$ .

Answer

64

Type 2common2 practice Q

Proper and non-empty subsets

How to spot it:

The word 'proper' (drop the set itself) or 'non-empty' (drop the empty set) is attached to subsets.

2^n - 1,\ \text{non-empty proper} = 2^n - 2

Method

Compute $2^n$ .
Subtract 1 for proper, 1 for non-empty.
Subtract 2 when both words appear.

Why it works:

Proper removes the full set and non-empty removes the empty set from the $2^n$ total.

Try this

A set has 5 elements. How many non-empty proper subsets does it have?

Show solution

Subsets: $2^5 = 32$ .
Non-empty proper: $32 - 2 = 30$ .

Answer

30

Type 3common2 practice Q

Subsets of a fixed size or containing given elements

How to spot it:

'Exactly 2 elements', 'containing a particular element', or 'at least one of two given elements'.

^{n}C_r

Method

Fixed size r: answer $^{n}C_r$ .
Must contain a set of k elements: choose the rest from $n - k$ .
At least one of some elements: $2^n$ minus the subsets avoiding all of them.

Why it works:

Fixing or banning elements just shrinks the pool that the remaining choice works on.

Try this

A set has 6 elements. How many subsets have exactly 2 elements?

Show solution

Count size-2 subsets: $^{6}C_2$ .
$\dfrac{6 \times 5}{2} = 15$ .

Answer

15

Type 4very common4 practice Q

Union of two sets and the difference region

How to spot it:

$n(A \cup B)$ or $n(A - B)$ is asked from counts of $A$ , $B$ and $A \cap B$ .

n(A \cup B) = n(A) + n(B) - n(A \cap B);\quad n(A - B) = n(A) - n(A \cap B)

Method

Write the union formula (or the difference formula).
Substitute the three counts.
Subtract the overlap for a union; subtract it from $n(A)$ for a difference.

Why it works:

The plain sum double-counts the overlap, so it must be removed once.

Try this

If $n(A) = 21$ , $n(B) = 17$ and $n(A \cap B) = 8$ , find $n(A \cup B)$ .

Show solution

$21 + 17 - 8$ .
$= 30$ .

Answer

30

Type 5common2 practice Q

De Morgan complement count

How to spot it:

The ask uses complements like $n(A' \cap B')$ or words like 'outside both A and B'.

n(A' \cap B') = n(U) - n(A \cup B)

Method

Convert the complement expression with De Morgan.
Find $n(A \cup B)$ from the counts.
Subtract the union from $n(U)$ .

Why it works:

'Not (A or B)' equals 'not A and not B', so the complement of the union is the answer.

Try this

In a universal set of 50 elements, $n(A) = 30$ , $n(B) = 24$ and $n(A \cap B) = 10$ . Find $n(A' \cap B')$ .

Show solution

$A' \cap B' = (A \cup B)'$ .
Union: $30 + 24 - 10 = 44$ .
$50 - 44 = 6$ .

Answer

6

08

Formula sheet

Number of subsets

2^n

A set with $n$ elements; the power set has $2^n$ elements.

Proper subsets

2^n - 1

All subsets except the set itself.

Non-empty subsets

2^n - 1

All subsets except the empty set.

Non-empty proper subsets

2^n - 2

Drops both the empty set and the full set.

Subsets of size r

^{n}C_r

Order does not matter inside a subset.

Union of two sets

n(A \cup B) = n(A) + n(B) - n(A \cap B)

Subtract the overlap counted twice.

De Morgan's laws

(A \cup B)' = A' \cap B',\quad (A \cap B)' = A' \cup B'

A complement swaps the join sign.

09

Shortcuts that save time

⚡ Know powers of 2 by heart

Subset questions are pure powers of two. Learn $2, 4, 8, \ldots, 512$ and every variant (proper, non-empty) becomes a one-step subtraction.

Example

A set has 9 elements. How many subsets does it have?

Show solution

Subsets $= 2^9$ .
$= 512$ .

Answer

512

⚡ Count what you do not want

For 'at least one of these elements' or 'non-empty proper', count the unwanted subsets and subtract from $2^n$ .

Example

A set has 6 elements. How many non-empty proper subsets does it have?

Show solution

Subsets: $2^6 = 64$ .
Drop the empty and the full set: $64 - 2 = 62$ .

Answer

62

⚡ De Morgan turns the question into one union

$n(A' \cap B')$ looks scary. De Morgan converts it to $n(U) - n(A \cup B)$ , a two-line count.

Example

In a universal set of 80 elements, $n(A) = 45$ , $n(B) = 35$ and $n(A \cap B) = 15$ . Find $n(A' \cap B')$ .

Show solution

Union: $45 + 35 - 15 = 65$ .
Complement: $80 - 65 = 15$ .

Answer

15

10

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Using $2^n$ where a smaller count is asked — proper subsets are $2^n - 1$ and non-empty proper subsets are $2^n - 2$ .

Mistake 02

Adding $n(A)$ and $n(B)$ without subtracting the overlap — the common elements get counted twice.

Mistake 03

Reading $n(A - B)$ as $n(A) - n(B)$ — subtract the overlap $n(A \cap B)$ , not the whole of $n(B)$ .

Mistake 04

Mixing up De Morgan — $(A \cup B)'$ becomes $A' \cap B'$ , not $A' \cup B'$ .

Mistake 05

Forgetting the empty set when listing subsets — $\emptyset$ is a subset of every set.

Mistake 06

Treating equivalent sets as equal — equal needs the same elements; equivalent only the same count.

11

Quick revision

Read this the night before the exam.

Subsets of an $n$ -element set: $2^n$ ; power set has $2^n$ elements.
Proper subsets $2^n - 1$ ; non-empty proper $2^n - 2$ .
Fixed size $r$ : $^{n}C_r$ subsets.
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$ .
$n(A - B) = n(A) - n(A \cap B)$ .
De Morgan: $(A \cup B)' = A' \cap B'$ , $(A \cap B)' = A' \cup B'$ .
$n(A' \cap B') = n(U) - n(A \cup B)$ .

12

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.