What is set notation in the context of Cambridge IGCSE Mathematics?

Set notation is a mathematical language used to define and describe collections of objects, known as sets. In the Cambridge IGCSE Mathematics curriculum, students learn to use symbols like {}, ∈, ∉, ⊆, and ∪ to represent and manipulate sets effectively.

How do you represent a set using set notation?

In set notation, a set is typically represented by listing its elements within curly braces. For example, the set of vowels in the English alphabet can be written as {a, e, i, o, u}. Alternatively, a set can be defined using a rule, such as {x | x is a positive even number}, which describes the set of all positive even numbers.

What is the difference between ∈ and ⊆ in set notation?

In set notation, the symbol ∈ denotes 'is an element of,' indicating that a particular object belongs to a set. For example, if A = {1, 2, 3}, then 2 ∈ A. On the other hand, ⊆ denotes 'is a subset of,' meaning all elements of one set are contained within another set. For instance, if B = {1, 2}, then B ⊆ A.

How do you express the union and intersection of sets in set notation?

The union of two sets, represented by the symbol ∪, is a set containing all elements from both sets. For example, if A = {1, 2} and B = {2, 3}, then A ∪ B = {1, 2, 3}. The intersection of two sets, represented by the symbol ∩, is a set containing only the elements common to both sets. For the same sets A and B, A ∩ B = {2}.

What is the significance of the empty set in set notation?

The empty set, denoted by {} or ∅, is a fundamental concept in set notation representing a set with no elements. It is significant because it serves as the identity element for the operation of union and is a subset of every set. Understanding the empty set is crucial for solving problems involving set operations in the Cambridge IGCSE Mathematics curriculum.

Why is "Confusing $\cup$ with $\cap$" a common mistake in Set Notation?

Why it happens: The symbols look similar. How to avoid it: is the U for Union ('OR'). is intersection ('AND'). Source: 0580/42 — recurring

Why is "Double-counting in $n(A \cup B)$" a common mistake in Set Notation?

Why it happens: Adding n(A) + n(B) without subtracting overlap. How to avoid it: Always subtract n(A B) to avoid counting shared elements twice.

Why is "Writing $\{0\}$ for the empty set" a common mistake in Set Notation?

Why it happens: Confusing 'no elements' with 'contains zero'. How to avoid it: = \\,\ has NO elements. \0\ has ONE element (which is the number 0).

Why is "Computing $A'$ without referencing $\mathcal{E}$" a common mistake in Set Notation?

Why it happens: Forgetting that complement depends on the universal set. How to avoid it: A' = E A. Always check what E has been defined as.

Cambridge IGCSE Mathematics (0580)

Set Notation

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Short Notes - Set Notation

Detailed Study Notes

Detailed notes on Sets for Cambridge IGCSE Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

Set Notation — Cambridge IGCSE 0580 Maths Extended (2026)

Sets, elements, subsets, union, intersection, complement, set-builder notation. Cambridge expects FLUENCY with the symbols — they're language, not afterthought.

At a glance

Set: a collection of distinct objects (elements). $A = \{1, 2, 3\}$ .
Element: $x \in A$ means $x$ is in $A$ . $x \notin A$ means it isn't.
Subset: $A \subseteq B$ — every element of $A$ is in $B$ . $A \subset B$ is a PROPER subset.
Union: $A \cup B$ — elements in $A$ OR $B$ (or both).
Intersection: $A \cap B$ — elements in BOTH $A$ AND $B$ .
Complement: $A'$ — elements in the universal set $\xi$ but NOT in $A$ .
Empty set: $\varnothing$ or $\{\}$ — no elements.
$n(A)$ : the number of elements in $A$ .

What you’ll learn

Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).

E9.1 — Use set language, notation and Venn diagrams to describe sets.
E9.1 — Use the notation $n(A)$ , $\xi$ , $\varnothing$ , $A'$ , $A \cup B$ , $A \cap B$ , $A \subseteq B$ , $A \subset B$ .

Sets, elements, and the universal set

A set is an unordered collection of distinct elements. Cambridge uses curly braces.

Set. A collection of distinct elements. List inside curly braces: $A = \{2, 4, 6, 8\}.$

Element of. $x \in A$ reads " $x$ is an element of $A$ " (i.e. $x$ is in the set).

$2 \in A, \quad 5 \notin A.$

Universal set. $\xi$ (Greek letter "xi") is the set of all elements under consideration. Often given at the start of a question, e.g. $\xi = \{1, 2, 3, \ldots, 10\}$ .

Empty set. $\varnothing$ or $\{\}$ has no elements. $n(\varnothing) = 0$ .

Cardinality. $n(A)$ is the number of elements in $A$ .

Worked. $\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ , $A = \{1, 3, 5, 7, 9\}$ .

$n(A) = 5$ .
$7 \in A$ , $8 \notin A$ .

Symbol	Meaning
$\in$ / $\notin$	is / is not an element of
$\xi$	universal set
$\varnothing$	empty set
$n(A)$	number of elements in $A$
$\subseteq$ / $\subset$	subset / proper subset
$\cup$ / $\cap$	union / intersection
$A'$	complement of $A$

Sets use curly braces.
$x \in A$ vs $x \notin A$ .
$\xi$ = universal set.
$\varnothing$ = empty set.
$n(A)$ = number of elements.

Subsets

$A \subseteq B$ : every element of $A$ is in $B$ . Proper subset: $A \subset B$ ( $A \ne B$ ).

Subset. $A \subseteq B$ means every element of $A$ is also an element of $B$ .

Proper subset. $A \subset B$ means $A \subseteq B$ AND $A \ne B$ (strictly fewer elements).

Every element of A lies inside B, so A ⊆ B; because A ≠ B it is also a proper subset A ⊂ B.

Worked. $A = \{1, 2\}$ , $B = \{1, 2, 3, 4\}$ , $C = \{1, 2\}$ .

$A \subseteq B$ ✓ (every element of $A$ is in $B$ ).
$A \subset B$ ✓ (proper — $A \ne B$ ).
$A \subseteq C$ ✓ (sets are equal).
$A \subset C$ ✗ (not proper — they're equal).

Special cases.

$\varnothing \subseteq A$ for every set $A$ .
$A \subseteq A$ for every set $A$ .
A set with $n$ elements has $2^{n}$ subsets.

Worked. How many subsets does $\{a, b, c\}$ have?

$2^{3} = 8$ . They are: $\varnothing, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\}$ .

$A \subseteq B$ : $A$ contained in (or equal to) $B$ .
$A \subset B$ : proper subset ( $A \ne B$ ).
$\varnothing \subseteq A$ always.
$n$ elements → $2^{n}$ subsets.

Union, intersection, complement

Union: OR. Intersection: AND. Complement: NOT.

Union. $A \cup B$ contains elements in $A$ OR $B$ (or both). $A \cup B = \{x : x \in A \text{ or } x \in B\}.$

Intersection. $A \cap B$ contains elements in BOTH $A$ AND $B$ . $A \cap B = \{x : x \in A \text{ and } x \in B\}.$

Complement. $A'$ contains elements in $\xi$ NOT in $A$ . $A' = \{x \in \xi : x \notin A\}.$

Worked. $\xi = \{1, 2, \ldots, 10\}$ , $A = \{1, 3, 5, 7, 9\}$ (odd), $B = \{2, 3, 5, 7\}$ (primes $\le 10$ ).

$A \cup B = \{1, 2, 3, 5, 7, 9\}$ .
$A \cap B = \{3, 5, 7\}$ .
$A' = \{2, 4, 6, 8, 10\}$ (even numbers in $\xi$ ).
$(A \cup B)' = \{4, 6, 8, 10\}$ (numbers in $\xi$ that are neither odd nor prime).

The overlap holds A ∩ B; the region outside both circles holds (A ∪ B)′.

Union shades everything in either circle; intersection shades only the overlap.

Union ( $\cup$ ): OR.
Intersection ( $\cap$ ): AND.
Complement ( $A'$ ): NOT (within $\xi$ ).
Always specify $\xi$ when computing complements.

Set-builder notation

Define a set by a property: $\{x : \text{condition on } x\}$ .

Set-builder. Instead of listing elements, describe them by a property: $A = \{x : x \text{ is an integer and } 1 \le x \le 10\}.$

The colon ":" reads "such that". (Some books use a vertical bar |.)

Worked translations.

$\{x : x \in \mathbb{Z}, x^{2} < 20\}$ → integers with $x^{2} < 20$ → $\{-4, -3, -2, -1, 0, 1, 2, 3, 4\}$ .
$\{x : x \text{ is a prime}, x < 15\}$ → $\{2, 3, 5, 7, 11, 13\}$ .

Common number sets.

$\mathbb{N}$ : natural numbers $\{0, 1, 2, 3, \ldots\}$ (sometimes from $1$ ).
$\mathbb{Z}$ : integers $\{\ldots, -2, -1, 0, 1, 2, \ldots\}$ .
$\mathbb{Q}$ : rationals.
$\mathbb{R}$ : reals.

Each number set is a subset of the next: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ.

Cambridge tip. Cambridge usually defines $\xi$ explicitly at the start of the question, then uses set-builder or list notation for individual sets. Match their exact notation in your answer.

Set-builder: $\{x : \text{condition}\}$ .
Colon means 'such that'.
Memorise $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$ .

Quick recap

$x \in A$ vs $x \notin A$ .
$\xi$ universal set; $\varnothing$ empty set; $n(A)$ count.
$A \subseteq B$ , $A \subset B$ (proper).
Union $\cup$ (OR), Intersection $\cap$ (AND), Complement $A'$ (NOT).
Set-builder: $\{x : \text{condition}\}$ .
Number sets: $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$ .

Memorise this

Verbatim phrases and definitions Cambridge mark schemes credit.

Set — a collection of distinct elements.
Universal set $\xi$ — the set of all elements under consideration.
Empty set $\varnothing$ — set with no elements.
Subset $\subseteq$ — every element of one set is in another.
Proper subset $\subset$ — subset and not equal.
Union $\cup$ — elements in either set.
Intersection $\cap$ — elements in both sets.
Complement $A'$ — elements in $\xi$ but not in $A$ .

How it’s examined

Set notation underpins every Venn diagram question — appears every Paper 2 and Paper 4 (typically 4-6 marks combined). Cambridge often gives a Venn diagram and asks you to write a region in set notation, or vice versa. Examiner reports flag confusing $\cup$ and $\cap$ and forgetting to take the complement WITHIN the universal set.

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E9.1); 0580/22 May/Jun 2024 — Q14 (set notation); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

Take this whole topic with you

Step-by-step worked examples — Set Notation

Step-by-step solutions to past-paper-style questions on set notation, written exactly the way a tutor would explain them at the board.

Question type:

Question patterns to master — Set Notation

Almost every set notation exam question is one of these shapes. Learn to spot each one and you will always know how to start.

Identify & classify

Recognise it by

The question gives a rule or a listed set and asks you to list the elements that satisfy a condition, or to write a given set in set-builder notation.

How to approach it

Work straight from the defining property: test each candidate value against the membership condition, or spot the pattern (even numbers as $2k$ , primes, multiples) and translate it into $\{x : \dots\}$ form.

Common trap

Examiner reports flag using everyday English instead of the IGCSE symbols, and including or omitting boundary values — read whether the condition uses $\le$ or $<$ .

Direct calculation

Recognise it by

A single instruction — find $A \cup B$ , find $A'$ , find $n(A \cup B)$ — on explicitly listed sets.

How to approach it

Apply the matching operation directly: collect elements for union/intersection/complement, or use $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ for a count. Always check what $\mathcal{E}$ is before taking a complement.

Common trap

Examiner reports flag swapping $\cup$ and $\cap$ , double-counting in $n(A \cup B)$ , and computing a complement without reference to the defined universal set.

Graph or diagram

Recognise it by

A Venn diagram is shown and you must shade a region, or read a shaded region and express it in set notation.

How to approach it

Decode the notation region by region — $\cap$ is the overlap, $\cup$ is everything touched, the dash is everything outside — then shade or name exactly those regions, including the area outside all circles when relevant.

Common trap

Examiner reports flag shading only the outside-all region for $(A \cap B)'$ and forgetting the single-set parts, and writing answers with $\setminus$ instead of the $\cup$ , $\cap$ , $'$ symbols.

Show that / prove

Recognise it by

The words verify, show that or confirm — typically a set identity such as a De Morgan law, with the result already stated.

How to approach it

Evaluate each side of the identity independently by listing its elements, then state explicitly that the two element lists are identical.

Common trap

Examiner reports flag sign slips — computing $A' \cap B'$ when $A' \cup B'$ is required — and stopping without an explicit concluding statement that both sides match.

1List elements of a set
CoreIdentify & classify• list
Question
Let $A = \{x : x \text{ is a prime number less than } 12\}$ . List the elements.
Step-by-step solution
1. Step 1
  Identify primes below $12$ .
  $A = \{2, 3, 5, 7, 11\}$
Answer
$A = \{2, 3, 5, 7, 11\}$
2Union and intersection
CoreDirect calculation• Adapted from 0580/22 May/Jun 2024 Q11• union, intersection
Question
$A = \{1, 2, 3, 4\}$ , $B = \{3, 4, 5, 6\}$ . Find $A \cup B$ and $A \cap B$ .
Step-by-step solution
1. Step 1
  Union = elements in $A$ OR $B$ (no duplicates).
  $A \cup B = \{1, 2, 3, 4, 5, 6\}$
2. Step 2
  Intersection = elements in $A$ AND $B$ .
  $A \cap B = \{3, 4\}$
Answer
$A \cup B = \{1, 2, 3, 4, 5, 6\},\ A \cap B = \{3, 4\}$
3Complement of a set
ExtendedDirect calculation• complement
Question
$\mathcal{E} = \{1, 2, 3, 4, 5, 6, 7, 8\}$ and $A = \{2, 4, 6, 8\}$ . Find $A'$ .
Step-by-step solution
1. Step 1
  $A'$ = elements of $\mathcal{E}$ NOT in $A$ .
  $A' = \{1, 3, 5, 7\}$
Answer
$A' = \{1, 3, 5, 7\}$
4Number of elements
ExtendedDirect calculation• cardinality
Question
$A = \{a, b, c, d\}$ and $B = \{c, d, e\}$ . Find $n(A \cup B)$ .
Step-by-step solution
1. Step 1
  Use $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ .
  $n(A \cup B) = 4 + 3 - 2 = 5$
Answer
$5$
5Set-builder notation
CoreIdentify & classify• set builder
Question
Write the set $\{2, 4, 6, 8, 10\}$ using set-builder notation.
Step-by-step solution
1. Step 1
  Identify the pattern: positive even integers up to and including $10$ .
2. Step 2
  Use 'such that' notation. Even numbers can be written as $2k$ for $k \in \mathbb{Z}$ .
  $\{x : x = 2k,\ k \in \mathbb{Z},\ 1 \le k \le 5\}$
3. Step 3
  Or, more directly: $\{x : x \in \mathbb{Z}^{+},\ x \text{ even},\ x \le 10\}$ .
Answer
$\{x : x = 2k,\ k \in \mathbb{Z},\ 1 \le k \le 5\}$
6Shade $(A \cap B)'$ on a Venn diagram
ExtendedGraph or diagram• Adapted from 0580/22 Oct/Nov 2023 Q12• shading, complement
Question
On a Venn diagram of two overlapping sets $A$ and $B$ inside universal set $\mathcal{E}$ , describe the region $(A \cap B)'$ .
Step-by-step solution
1. Step 1
  $A \cap B$ is the central lens — elements in both.
2. Step 2
  The complement is everything EXCEPT that lens: $A$ only, $B$ only, and the region outside both.
Answer
Shade everywhere except the lens — i.e. $A$ only, $B$ only, and the area outside both circles.
Examiner tip
The examiner report flags candidates who shade only the area outside both circles, forgetting that 'NOT in both' includes the single-set regions too.
7Apply $(A \cup B)'$
ExtendedDirect calculation• complement, union
Question
$\mathcal{E} = \{1, 2, \dots, 12\}$ , $A = \{2, 4, 6, 8, 10, 12\}$ (even), $B = \{3, 6, 9, 12\}$ (multiples of $3$ ). List the elements of $(A \cup B)'$ .
Step-by-step solution
1. Step 1
  Find the union first.
  $A \cup B = \{2, 3, 4, 6, 8, 9, 10, 12\}$
2. Step 2
  The complement is $\mathcal{E} \setminus (A \cup B)$ .
  $(A \cup B)' = \{1, 5, 7, 11\}$
Answer
$\{1, 5, 7, 11\}$
8Identify a region using set notation (Challenge)
ChallengeGraph or diagram• Adapted from 0580/42 May/Jun 2024 Q14• region, challenge
Question
On a Venn diagram of three overlapping sets $A$ , $B$ and $C$ , the region containing elements in $A$ AND in $C$ but NOT in $B$ is shaded. Express this region using set notation.
Step-by-step solution
1. Step 1
  Translate the description: 'in $A$ ' and 'in $C$ ' is $A \cap C$ .
2. Step 2
  'Not in $B$ ' is the complement $B'$ .
3. Step 3
  Combine.
  $A \cap C \cap B'$
Answer
$A \cap B' \cap C$ (equivalently $A \cap C \cap B'$ ).
Examiner tip
The examiner report consistently flags candidates writing $(A \cap C) \setminus B$ — which is correct but not in IGCSE notation. Stick to the $\cup$ , $\cap$ , and $'$ symbols.
9Apply De Morgan's law (Challenge)
ChallengeShow that / prove• de morgan, challenge
Question
$\mathcal{E} = \{1, 2, \dots, 10\}$ , $A = \{1, 2, 3, 4, 5\}$ , $B = \{4, 5, 6, 7, 8\}$ . Verify by listing that $(A \cap B)' = A' \cup B'$ .
Step-by-step solution
1. Step 1
  Left side.
  $A \cap B = \{4, 5\} \Rightarrow (A \cap B)' = \{1, 2, 3, 6, 7, 8, 9, 10\}$
2. Step 2
  Right side: complements first.
  $A' = \{6, 7, 8, 9, 10\};\ B' = \{1, 2, 3, 9, 10\}$
3. Step 3
  Union.
  $A' \cup B' = \{1, 2, 3, 6, 7, 8, 9, 10\}$
4. Step 4
  Both sides equal — De Morgan's law confirmed.
Answer
Both sides $= \{1, 2, 3, 6, 7, 8, 9, 10\}$ .
Examiner tip
De Morgan's laws are testable but rarely named explicitly. The examiner report flags students who try to compute $A' \cap B'$ on the right side instead of $A' \cup B'$ — sign error.

Key Formulae — Set Notation

The formulae you need to memorise for set notation on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.

Inclusion-exclusion (two sets)
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
$n(X)$
number of elements in set $X$
When to use
Counting elements of a union without listing.
Complement rule
$n(A') = n(\mathcal{E}) - n(A)$
$\mathcal{E}$
universal set
When to use
Find the size of a complement.

Key Definitions and Keywords — Set Notation

Definitions to memorise and the exact keywords mark schemes credit for set notation answers — sharpened from recent examiner reports for the 2026 0580 sitting.

Set
Examiner keyword
A collection of distinct elements, written inside braces $\{\,\}$ .
Universal set ( $\mathcal{E}$ )
Examiner keyword
The set containing every element under consideration.
Empty set ( $\emptyset$ or $\{\,\}$ )
Examiner keyword
The set with no elements.
Subset ( $A \subseteq B$ )
Examiner keyword
Every element of $A$ is also in $B$ . Strict (proper) subset uses $\subset$ .
Union ( $A \cup B$ )
Examiner keyword
All elements that are in $A$ OR $B$ (or both).
Intersection ( $A \cap B$ )
Examiner keyword
Elements that are in BOTH $A$ and $B$ .
Complement ( $A'$ )
Examiner keyword
Elements of the universal set NOT in $A$ .

Common Mistakes and Misconceptions — Set Notation

The traps other students keep falling into on set notation questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.

✕Confusing $\cup$ with $\cap$
0580/42 — recurring
Why it happens
The symbols look similar.
How to avoid it
$\cup$ is the U for Union ('OR'). $\cap$ is intersection ('AND').
✕Double-counting in $n(A \cup B)$
Why it happens
Adding $n(A) + n(B)$ without subtracting overlap.
How to avoid it
Always subtract $n(A \cap B)$ to avoid counting shared elements twice.
✕Writing $\{0\}$ for the empty set
Why it happens
Confusing 'no elements' with 'contains zero'.
How to avoid it
$\emptyset = \{\,\}$ has NO elements. $\{0\}$ has ONE element (which is the number 0).
✕Computing $A'$ without referencing $\mathcal{E}$
Why it happens
Forgetting that complement depends on the universal set.
How to avoid it
$A'$ = $\mathcal{E} \setminus A$ . Always check what $\mathcal{E}$ has been defined as.

Set Notation — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Symbol

Meaning

\in

\notin

is / is not an element of

\xi

universal set

\varnothing

empty set

n(A)

number of elements in

A

\subseteq

\subset

subset / proper subset

\cup

\cap

union / intersection

A'

complement of

A

Set-builder. Instead of listing elements, describe them by a property: $A = \{x : x \text{ is an integer and } 1 \le x \le 10\}.$

The colon ":" reads "such that". (Some books use a vertical bar |.)

Worked translations.

$\{x : x \in \mathbb{Z}, x^{2} < 20\}$ → integers with $x^{2} < 20$ → $\{-4, -3, -2, -1, 0, 1, 2, 3, 4\}$ .
$\{x : x \text{ is a prime}, x < 15\}$ → $\{2, 3, 5, 7, 11, 13\}$ .

Common number sets.

$\mathbb{N}$ : natural numbers $\{0, 1, 2, 3, \ldots\}$ (sometimes from $1$ ).
$\mathbb{Z}$ : integers $\{\ldots, -2, -1, 0, 1, 2, \ldots\}$ .
$\mathbb{Q}$ : rationals.
$\mathbb{R}$ : reals.

Each number set is a subset of the next: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ.

Cambridge tip. Cambridge usually defines $\xi$ explicitly at the start of the question, then uses set-builder or list notation for individual sets. Match their exact notation in your answer.

Set Notation

Short Notes - Set Notation

Detailed Study Notes

At a glance

What you’ll learn

Sets, elements, and the universal set

Subsets

Union, intersection, complement

Set-builder notation

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

Question patterns to master — Set Notation

1List elements of a set

2Union and intersection

3Complement of a set

4Number of elements

5Set-builder notation

6Shade (A∩B)′(A \cap B)'(A∩B)′ on a Venn diagram

7Apply (A∪B)′(A \cup B)'(A∪B)′

8Identify a region using set notation (Challenge)

9Apply De Morgan's law (Challenge)

Inclusion-exclusion (two sets)

Complement rule

Set

Universal set (E\mathcal{E}E)

Empty set (∅\emptyset∅ or { }\{\,\}{})

Subset (A⊆BA \subseteq BA⊆B)

Union (A∪BA \cup BA∪B)

Intersection (A∩BA \cap BA∩B)

Complement (A′A'A′)

✕Confusing ∪\cup∪ with ∩\cap∩

✕Double-counting in n(A∪B)n(A \cup B)n(A∪B)

✕Writing {0}\{0\}{0} for the empty set

✕Computing A′A'A′ without referencing E\mathcal{E}E

Set Notation — frequently asked questions

Set Notation

Short Notes - Set Notation

Detailed Study Notes

At a glance

What you’ll learn

Sets, elements, and the universal set

Subsets

Union, intersection, complement

Set-builder notation

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

Question patterns to master — Set Notation

1List elements of a set

2Union and intersection

3Complement of a set

4Number of elements

5Set-builder notation

6Shade (A∩B)′(A \cap B)'(A∩B)′ on a Venn diagram

7Apply (A∪B)′(A \cup B)'(A∪B)′

8Identify a region using set notation (Challenge)

9Apply De Morgan's law (Challenge)

Inclusion-exclusion (two sets)

Complement rule

Set

Universal set (E\mathcal{E}E)

Empty set (∅\emptyset∅ or { }\{\,\}{})

Subset (A⊆BA \subseteq BA⊆B)

Union (A∪BA \cup BA∪B)

Intersection (A∩BA \cap BA∩B)

Complement (A′A'A′)

✕Confusing ∪\cup∪ with ∩\cap∩

✕Double-counting in n(A∪B)n(A \cup B)n(A∪B)

✕Writing {0}\{0\}{0} for the empty set

✕Computing A′A'A′ without referencing E\mathcal{E}E

Set Notation — frequently asked questions

6Shade $(A \cap B)'$ on a Venn diagram

7Apply $(A \cup B)'$

Universal set ( $\mathcal{E}$ )

Empty set ( $\emptyset$ or $\{\,\}$ )

Subset ( $A \subseteq B$ )

Union ( $A \cup B$ )

Intersection ( $A \cap B$ )

Complement ( $A'$ )

✕Confusing $\cup$ with $\cap$

✕Double-counting in $n(A \cup B)$

✕Writing $\{0\}$ for the empty set

✕Computing $A'$ without referencing $\mathcal{E}$

6Shade $(A \cap B)'$ on a Venn diagram

7Apply $(A \cup B)'$

Universal set ( $\mathcal{E}$ )

Empty set ( $\emptyset$ or $\{\,\}$ )

Subset ( $A \subseteq B$ )

Union ( $A \cup B$ )

Intersection ( $A \cap B$ )

Complement ( $A'$ )

✕Confusing $\cup$ with $\cap$

✕Double-counting in $n(A \cup B)$

✕Writing $\{0\}$ for the empty set

✕Computing $A'$ without referencing $\mathcal{E}$