What is a Venn Diagram and how is it used in Mathematics?

A Venn Diagram is a visual tool used in Mathematics to illustrate the relationships between different sets. It consists of overlapping circles, each representing a set, where the overlap shows the common elements. Venn Diagrams are particularly useful in solving problems related to union, intersection, and complement of sets, which are key concepts in the Cambridge IGCSE Mathematics curriculum.

How do you represent the union and intersection of sets using a Venn Diagram?

In a Venn Diagram, the union of sets is represented by the area covered by all the circles involved, indicating all elements that belong to any of the sets. The intersection is shown by the overlapping area of the circles, representing elements common to all sets. Understanding these concepts is crucial for solving set-related problems in the Cambridge IGCSE Mathematics syllabus.

What are some common applications of Venn Diagrams in solving mathematical problems?

Venn Diagrams are commonly used to solve problems involving set operations such as finding the union, intersection, and difference of sets. They are also helpful in visualizing and solving problems related to probability, logic, and data organization, which are important topics in the Cambridge IGCSE Mathematics curriculum.

How can Venn Diagrams help in understanding the concept of set complements?

A Venn Diagram helps in visualizing the complement of a set by highlighting the area outside the circle representing the set, within the universal set. This visual representation makes it easier to understand which elements are not included in the set, a concept that is essential in set theory as taught in the Cambridge IGCSE Mathematics course.

What are some tips for drawing accurate Venn Diagrams for complex set problems?

To draw accurate Venn Diagrams for complex set problems, start by clearly defining each set and its elements. Use distinct circles for each set and ensure they overlap correctly to represent intersections. Label each section of the diagram to avoid confusion. Practicing with different set operations and problems, as outlined in the Cambridge IGCSE Mathematics curriculum, can also enhance your skills in using Venn Diagrams effectively.

Why is "Filling outer regions before the centre" a common mistake in Venn Diagrams?

Why it happens: Reading questions left-to-right. How to avoid it: Always start at A B C and work outwards. Source: 0580/42 — every series

Why is "Putting the full pairwise count in the lens region" a common mistake in Venn Diagrams?

Why it happens: n(M P) = 7 but the LENS region shows ONLY M P minus the centre. How to avoid it: Pairwise lens region = n(A B) - n(A B C).

Why is "Confusing '$A$ only' with '$A$'" a common mistake in Venn Diagrams?

Why it happens: A includes overlapping regions; 'A only' excludes them. How to avoid it: Read carefully — 'students who study Maths' includes overlaps; 'students who study ONLY Maths' excludes them.

Why is "Forgetting the universal-set region" a common mistake in Venn Diagrams?

Why it happens: Focusing on the named sets. How to avoid it: After filling the circles, check that the rectangle adds up to n(E).

SetsCambridge IGCSE Maths 0580 Extended

Venn Diagrams

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Venn Diagrams — Cambridge IGCSE 0580 Maths Extended (2026)

Two- and three-set Venn diagrams. Shade regions to match set expressions, fill in numbers from word problems, and compute counts and probabilities.

What you’ll learn

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

E9.2 — Use Venn diagrams to represent sets and solve problems.

Two-set Venn diagrams

Rectangle $\xi$ with two overlapping circles. Four regions to label.

Anatomy. A two-set Venn shows the universal set $\xi$ as a rectangle, with two circles $A$ and $B$ inside.

Four regions:

$A \cap B$ — both (overlap).
$A \cap B'$ — only $A$ (left crescent).
$A' \cap B$ — only $B$ (right crescent).
$(A \cup B)'$ — neither (outside both, inside the rectangle).

Two circles inside ξ create exactly four regions to label.

Worked. $\xi = \{1, 2, \ldots, 10\}$ . $A$ = multiples of $2$ . $B$ = multiples of $3$ .

$A \cap B = \{6\}$ (multiples of $6$ ).
$A \cap B' = \{2, 4, 8, 10\}$ .
$A' \cap B = \{3, 9\}$ .
$(A \cup B)' = \{1, 5, 7\}$ .

Inclusion-exclusion (counts). $n(A \cup B) = n(A) + n(B) - n(A \cap B).$ The intersection is subtracted because it would otherwise be counted twice.

4 regions in a 2-set Venn.
Inclusion-exclusion: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ .
Outside both circles is $(A \cup B)'$ .

Three-set Venn diagrams

Three overlapping circles → 8 regions. Always fill the centre first.

Eight regions in a 3-set Venn.

Method for word problems.

Identify $n(A \cap B \cap C)$ — the very centre — and fill it in.
Work OUTWARDS to the pairwise intersections (e.g. $n(A \cap B) - n(A \cap B \cap C)$ goes in the " $A$ and $B$ only" region).
Then to "only $A$ ", "only $B$ ", "only $C$ ".
Finally, the outside region: total minus everything inside.

Worked. In a class of $30$ students:

$18$ play football, $14$ play tennis, $10$ play cricket.
$7$ play football and tennis, $5$ play football and cricket, $4$ play tennis and cricket.
$2$ play all three.
How many play none?

Step 1. Centre: $n(F \cap T \cap C) = 2$ .

Step 2. Pairwise (after subtracting centre):

$F \cap T$ only: $7 - 2 = 5$ .
$F \cap C$ only: $5 - 2 = 3$ .
$T \cap C$ only: $4 - 2 = 2$ .

Step 3. Singles:

Only $F$ : $18 - (5 + 3 + 2) = 8$ .
Only $T$ : $14 - (5 + 2 + 2) = 5$ .
Only $C$ : $10 - (3 + 2 + 2) = 3$ .

Step 4. Total in any sport: $8 + 5 + 3 + 5 + 3 + 2 + 2 = 28$ .

Step 5. None: $30 - 28 = 2$ .

Centre filled first (2), then pair overlaps, then singles, then the outside — all eight counts add to 30.

8 regions in a 3-set Venn.
Fill the centre first, then pairwise, then singles, then outside.
Always subtract overlaps when working outwards.

Shading regions

Cambridge often asks 'shade the region representing $X$ '. Translate set notation step-by-step.

Method. Translate the set expression into a sequence of operations on the Venn:

Start with the simplest piece (a single set or its complement).
Apply union or intersection step-by-step.
Lightly shade with arrows or dashes — Cambridge accepts any clear shading.

Worked. Shade $A \cap B'$ .

$A$ = the left circle.
$B'$ = everything outside circle $B$ .
Intersection: parts of $A$ that are also outside $B$ → the left crescent ("only $A$ ").

A ∩ B′ is the part of A lying outside B — the left crescent only.

Worked. Shade $(A \cup B)' \cap C$ .

$A \cup B$ = both circles.
$(A \cup B)'$ = outside both $A$ and $B$ but inside $\xi$ .
$\cap C$ = also in $C$ .
Result: the part of $C$ that doesn't overlap with $A$ or $B$ — "only $C$ ".

Worked. Shade $A' \cup B$ .

$A'$ = outside $A$ (everything except the $A$ circle).
$B$ = the $B$ circle.
Union: everything outside $A$ TOGETHER with all of $B$ .
Result: everything except the "only $A$ " region.

Translate the expression piece by piece.
Build up from simplest sets / complements.
Apply union (OR) or intersection (AND) at each step.
Shade clearly — any consistent shading is fine.

Venn diagrams for probability

Once a Venn is filled with counts, probability is just count / total.

If $\xi$ contains $n$ equally likely outcomes: $P(A) = \dfrac{n(A)}{n(\xi)}.$

Worked. From the class of $30$ in the previous example, a student is chosen at random.

$P(\text{plays exactly one sport}) = \dfrac{8 + 5 + 3}{30} = \dfrac{16}{30} = \dfrac{8}{15}$ .
$P(\text{plays football and tennis}) = \dfrac{7}{30}$ (this counts the $5$ who play just F&T plus the $2$ who play all three).
$P(\text{plays no sport}) = \dfrac{2}{30} = \dfrac{1}{15}$ .

Tip. Cambridge probability questions often layer on top of a Venn — fill the diagram first, then read off counts as needed.

Probability $= n(\text{event}) / n(\xi)$ .
Read counts from the filled Venn.
'Plays exactly one' = sum of the three single regions.

How it’s examined

Venn diagram questions appear every Paper 2 (3-5 marks: shading or counting) and most Paper 4s (5-7 marks: a word problem followed by probability). Examiner reports flag the off-by-one error from forgetting to subtract overlaps and from confusing 'plays exactly one sport' with 'plays at least one sport'.

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E9.2); 0580/42 Oct/Nov 2024 — Q13 (3-set Venn word problem); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

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Step-by-step worked examples — Venn Diagrams

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

1Place elements in a two-set Venn diagram
Core• two set
Question
$\mathcal{E} = \{1,\dots,10\}$ , $A = \{2,4,6,8\}$ , $B = \{3,4,5,6\}$ . Place every element in the correct region.
Step-by-step solution
1. Step 1
  Intersection ( $A \cap B$ ) first.
  $A \cap B = \{4, 6\}$
2. Step 2
  $A$ only and $B$ only.
  $A \setminus B = \{2, 8\},\ B \setminus A = \{3, 5\}$
3. Step 3
  Outside both (the complement of $A \cup B$ ).
  $(A \cup B)' = \{1, 7, 9, 10\}$
Answer
$A \cap B = \{4,6\}$ ; $A$ only $= \{2,8\}$ ; $B$ only $= \{3,5\}$ ; outside $= \{1,7,9,10\}$ .
2Counting with a three-set Venn diagram
Extended• Adapted from 0580/42 May/Jun 2024 Q12• three set
Question
In a class of $30$ , $18$ study Maths, $14$ Physics, $10$ Chemistry. $7$ study Maths and Physics, $4$ Maths and Chemistry, $3$ Physics and Chemistry. $2$ study all three. Find how many study none.
Step-by-step solution
1. Step 1
  Start at the centre.
  $M \cap P \cap C = 2$
2. Step 2
  Pairwise minus centre.
  $M \cap P \text{ only} = 7 - 2 = 5;\ M \cap C \text{ only} = 4 - 2 = 2;\ P \cap C \text{ only} = 3 - 2 = 1$
3. Step 3
  Single-set 'only' regions.
  $M \text{ only} = 18 - 5 - 2 - 2 = 9;\ P \text{ only} = 14 - 5 - 1 - 2 = 6;\ C \text{ only} = 10 - 2 - 1 - 2 = 5$
4. Step 4
  Total inside any set.
  $9 + 6 + 5 + 5 + 2 + 1 + 2 = 30$
5. Step 5
  Outside any set.
  $30 - 30 = 0$
Answer
$0$ students study none.
Examiner tip
Always work from the CENTRE outwards — never from the perimeter inwards.
3Shade a region
Extended• shading
Question
Shade the region representing $A' \cap B$ .
Step-by-step solution
1. Step 1
  $A'$ = everything outside $A$ . Intersect with $B$ .
2. Step 2
  Result: the part of $B$ that lies OUTSIDE $A$ .
Answer
Shade the part of $B$ that does not overlap $A$ .
4Find $n(A \cap B)$ from totals
Core• Adapted from 0580/22 May/Jun 2023 Q11• intersection, two set
Question
$30$ students play sport. $20$ play football (F), $15$ play tennis (T), and every student plays at least one of the two. Find $n(F \cap T)$ .
Step-by-step solution
1. Step 1
  Since every student plays at least one, $n(F \cup T) = 30$ .
2. Step 2
  Use inclusion-exclusion: $n(F \cup T) = n(F) + n(T) - n(F \cap T)$ .
  $30 = 20 + 15 - n(F \cap T)$
3. Step 3
  Rearrange.
  $n(F \cap T) = 35 - 30 = 5$
Answer
$5$ students play both.
5Probability from a two-set Venn diagram
Extended• probability, two set
Question
In a class of $25$ , $14$ have a brother (B), $10$ have a sister (S), $5$ have both. A student is chosen at random. Find (a) $P(B \cap S)$ , (b) $P(B' \cap S')$ (has neither).
Step-by-step solution
1. Step 1
  $B$ only $= 14 - 5 = 9$ ; $S$ only $= 10 - 5 = 5$ ; both $= 5$ .
2. Step 2
  Inside the circles: $9 + 5 + 5 = 19$ . Outside: $25 - 19 = 6$ .
3. Step 3
  Probabilities.
  $P(B \cap S) = \dfrac{5}{25} = \dfrac{1}{5};\ P(B' \cap S') = \dfrac{6}{25}$
Answer
(a) $\dfrac{1}{5}$ (b) $\dfrac{6}{25}$
6Fill in a three-set Venn diagram
Extended• Adapted from 0580/42 Feb/Mar 2024 Q9• three set, fill
Question
Of $50$ tourists: $28$ visited London (L), $24$ visited Paris (P), $20$ visited Rome (R). $12$ visited L and P, $10$ visited P and R, $8$ visited L and R. $5$ visited all three. Find the number who visited none.
Step-by-step solution
1. Step 1
  Centre.
  $L \cap P \cap R = 5$
2. Step 2
  Pairwise lens regions (subtract the centre).
  $L \cap P \text{ only} = 12 - 5 = 7;\ P \cap R \text{ only} = 10 - 5 = 5;\ L \cap R \text{ only} = 8 - 5 = 3$
3. Step 3
  Single-set 'only' regions.
  $L \text{ only} = 28 - 7 - 3 - 5 = 13;\ P \text{ only} = 24 - 7 - 5 - 5 = 7;\ R \text{ only} = 20 - 5 - 3 - 5 = 7$
4. Step 4
  Total inside the circles.
  $13 + 7 + 7 + 7 + 5 + 3 + 5 = 47$
5. Step 5
  Outside.
  $50 - 47 = 3$
Answer
$3$ tourists visited none.
7Find a union using inclusion-exclusion
Extended• union, two set
Question
$n(\mathcal{E}) = 60$ , $n(A) = 32$ , $n(B) = 28$ , $n(A \cap B) = 14$ . Find (a) $n(A \cup B)$ , (b) $n((A \cup B)')$ .
Step-by-step solution
1. Step 1
  Inclusion-exclusion.
  $n(A \cup B) = 32 + 28 - 14 = 46$
2. Step 2
  Complement.
  $n((A \cup B)') = 60 - 46 = 14$
Answer
(a) $46$ (b) $14$
8Word-problem Venn (two subjects)
Extended• Adapted from 0580/42 May/Jun 2024 Q12• word problem, two set
Question
In a year group of $80$ students, $45$ study Spanish, $33$ study French, and $12$ study neither language. Find how many students study both Spanish and French.
Step-by-step solution
1. Step 1
  Students studying at least one language $= 80 - 12 = 68$ .
2. Step 2
  Use $n(S \cup F) = n(S) + n(F) - n(S \cap F)$ .
  $68 = 45 + 33 - n(S \cap F)$
3. Step 3
  Rearrange.
  $n(S \cap F) = 78 - 68 = 10$
Answer
$10$ students study both.
Examiner tip
The examiner report flags that candidates often forget the 'neither' group and use $n(S \cup F) = 80$ directly, getting the wrong intersection.
9Three-set Venn with unknowns (Challenge)
Challenge• Adapted from 0580/42 Oct/Nov 2024 Q12• three set, algebra, challenge
Question
In a survey of $100$ people: $50$ liked apples (A), $40$ bananas (B), $35$ cherries (C). $20$ liked both A and B, $15$ liked both A and C, $10$ liked both B and C. Let $x$ = number who liked all three. Given that $5$ people liked none, find $x$ .
Step-by-step solution
1. Step 1
  Number liking at least one fruit $= 100 - 5 = 95$ .
2. Step 2
  Apply three-set inclusion-exclusion.
  $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)$
3. Step 3
  Substitute.
  $95 = 50 + 40 + 35 - 20 - 15 - 10 + x$
4. Step 4
  Simplify.
  $95 = 80 + x \Rightarrow x = 15$
Answer
$x = 15$ people liked all three.
Examiner tip
The examiner report consistently flags sign errors in the three-set formula. Remember: pairwise terms are SUBTRACTED, but the triple intersection is ADDED back in.

Key Formulae — Venn Diagrams

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

Two-set counting
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
When to use
Two overlapping sets.
Three-set counting
$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)$
When to use
Three overlapping sets.

Key Definitions and Keywords — Venn Diagrams

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

Region
Examiner keyword
A specific zone of a Venn diagram representing an exact combination of memberships.
$A$ only
Examiner keyword
Elements in $A$ but not in any other named set. = $A \setminus (B \cup C)$ in a 3-set diagram.
Shading a region
Examiner keyword
Marking the area corresponding to a named set expression, e.g. $A' \cap B$ .

Common Mistakes and Misconceptions — Venn Diagrams

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

✕Filling outer regions before the centre
0580/42 — every series
Why it happens
Reading questions left-to-right.
How to avoid it
Always start at $A \cap B \cap C$ and work outwards.
✕Putting the full pairwise count in the lens region
Why it happens
$n(M \cap P) = 7$ but the LENS region shows ONLY $M \cap P$ minus the centre.
How to avoid it
Pairwise lens region = $n(A \cap B) - n(A \cap B \cap C)$ .
✕Confusing ' $A$ only' with ' $A$ '
Why it happens
$A$ includes overlapping regions; ' $A$ only' excludes them.
How to avoid it
Read carefully — 'students who study Maths' includes overlaps; 'students who study ONLY Maths' excludes them.
✕Forgetting the universal-set region
Why it happens
Focusing on the named sets.
How to avoid it
After filling the circles, check that the rectangle adds up to $n(\mathcal{E})$ .

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

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Venn Diagrams — frequently asked questions

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SetsCambridge IGCSE Maths 0580 Extended

Venn Diagrams

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Venn Diagrams — Cambridge IGCSE 0580 Maths Extended (2026)

Two- and three-set Venn diagrams. Shade regions to match set expressions, fill in numbers from word problems, and compute counts and probabilities.

What you’ll learn

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

E9.2 — Use Venn diagrams to represent sets and solve problems.

Two-set Venn diagrams

Rectangle $\xi$ with two overlapping circles. Four regions to label.

Anatomy. A two-set Venn shows the universal set $\xi$ as a rectangle, with two circles $A$ and $B$ inside.

Four regions:

$A \cap B$ — both (overlap).
$A \cap B'$ — only $A$ (left crescent).
$A' \cap B$ — only $B$ (right crescent).
$(A \cup B)'$ — neither (outside both, inside the rectangle).

Two circles inside ξ create exactly four regions to label.

Worked. $\xi = \{1, 2, \ldots, 10\}$ . $A$ = multiples of $2$ . $B$ = multiples of $3$ .

$A \cap B = \{6\}$ (multiples of $6$ ).
$A \cap B' = \{2, 4, 8, 10\}$ .
$A' \cap B = \{3, 9\}$ .
$(A \cup B)' = \{1, 5, 7\}$ .

Inclusion-exclusion (counts). $n(A \cup B) = n(A) + n(B) - n(A \cap B).$ The intersection is subtracted because it would otherwise be counted twice.

4 regions in a 2-set Venn.
Inclusion-exclusion: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ .
Outside both circles is $(A \cup B)'$ .

Three-set Venn diagrams

Three overlapping circles → 8 regions. Always fill the centre first.

Eight regions in a 3-set Venn.

Method for word problems.

Identify $n(A \cap B \cap C)$ — the very centre — and fill it in.
Work OUTWARDS to the pairwise intersections (e.g. $n(A \cap B) - n(A \cap B \cap C)$ goes in the " $A$ and $B$ only" region).
Then to "only $A$ ", "only $B$ ", "only $C$ ".
Finally, the outside region: total minus everything inside.

Worked. In a class of $30$ students:

$18$ play football, $14$ play tennis, $10$ play cricket.
$7$ play football and tennis, $5$ play football and cricket, $4$ play tennis and cricket.
$2$ play all three.
How many play none?

Step 1. Centre: $n(F \cap T \cap C) = 2$ .

Step 2. Pairwise (after subtracting centre):

$F \cap T$ only: $7 - 2 = 5$ .
$F \cap C$ only: $5 - 2 = 3$ .
$T \cap C$ only: $4 - 2 = 2$ .

Step 3. Singles:

Only $F$ : $18 - (5 + 3 + 2) = 8$ .
Only $T$ : $14 - (5 + 2 + 2) = 5$ .
Only $C$ : $10 - (3 + 2 + 2) = 3$ .

Step 4. Total in any sport: $8 + 5 + 3 + 5 + 3 + 2 + 2 = 28$ .

Step 5. None: $30 - 28 = 2$ .

Centre filled first (2), then pair overlaps, then singles, then the outside — all eight counts add to 30.

8 regions in a 3-set Venn.
Fill the centre first, then pairwise, then singles, then outside.
Always subtract overlaps when working outwards.

Shading regions

Cambridge often asks 'shade the region representing $X$ '. Translate set notation step-by-step.

Method. Translate the set expression into a sequence of operations on the Venn:

Start with the simplest piece (a single set or its complement).
Apply union or intersection step-by-step.
Lightly shade with arrows or dashes — Cambridge accepts any clear shading.

Worked. Shade $A \cap B'$ .

$A$ = the left circle.
$B'$ = everything outside circle $B$ .
Intersection: parts of $A$ that are also outside $B$ → the left crescent ("only $A$ ").

A ∩ B′ is the part of A lying outside B — the left crescent only.

Worked. Shade $(A \cup B)' \cap C$ .

$A \cup B$ = both circles.
$(A \cup B)'$ = outside both $A$ and $B$ but inside $\xi$ .
$\cap C$ = also in $C$ .
Result: the part of $C$ that doesn't overlap with $A$ or $B$ — "only $C$ ".

Worked. Shade $A' \cup B$ .

$A'$ = outside $A$ (everything except the $A$ circle).
$B$ = the $B$ circle.
Union: everything outside $A$ TOGETHER with all of $B$ .
Result: everything except the "only $A$ " region.

Translate the expression piece by piece.
Build up from simplest sets / complements.
Apply union (OR) or intersection (AND) at each step.
Shade clearly — any consistent shading is fine.

Venn diagrams for probability

Once a Venn is filled with counts, probability is just count / total.

If $\xi$ contains $n$ equally likely outcomes: $P(A) = \dfrac{n(A)}{n(\xi)}.$

Worked. From the class of $30$ in the previous example, a student is chosen at random.

$P(\text{plays exactly one sport}) = \dfrac{8 + 5 + 3}{30} = \dfrac{16}{30} = \dfrac{8}{15}$ .
$P(\text{plays football and tennis}) = \dfrac{7}{30}$ (this counts the $5$ who play just F&T plus the $2$ who play all three).
$P(\text{plays no sport}) = \dfrac{2}{30} = \dfrac{1}{15}$ .

Tip. Cambridge probability questions often layer on top of a Venn — fill the diagram first, then read off counts as needed.

Probability $= n(\text{event}) / n(\xi)$ .
Read counts from the filled Venn.
'Plays exactly one' = sum of the three single regions.

How it’s examined

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E9.2); 0580/42 Oct/Nov 2024 — Q13 (3-set Venn word problem); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

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Step-by-step worked examples — Venn Diagrams

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

1Place elements in a two-set Venn diagram
Core• two set
Question
$\mathcal{E} = \{1,\dots,10\}$ , $A = \{2,4,6,8\}$ , $B = \{3,4,5,6\}$ . Place every element in the correct region.
Step-by-step solution
1. Step 1
  Intersection ( $A \cap B$ ) first.
  $A \cap B = \{4, 6\}$
2. Step 2
  $A$ only and $B$ only.
  $A \setminus B = \{2, 8\},\ B \setminus A = \{3, 5\}$
3. Step 3
  Outside both (the complement of $A \cup B$ ).
  $(A \cup B)' = \{1, 7, 9, 10\}$
Answer
$A \cap B = \{4,6\}$ ; $A$ only $= \{2,8\}$ ; $B$ only $= \{3,5\}$ ; outside $= \{1,7,9,10\}$ .
2Counting with a three-set Venn diagram
Extended• Adapted from 0580/42 May/Jun 2024 Q12• three set
Question
In a class of $30$ , $18$ study Maths, $14$ Physics, $10$ Chemistry. $7$ study Maths and Physics, $4$ Maths and Chemistry, $3$ Physics and Chemistry. $2$ study all three. Find how many study none.
Step-by-step solution
1. Step 1
  Start at the centre.
  $M \cap P \cap C = 2$
2. Step 2
  Pairwise minus centre.
  $M \cap P \text{ only} = 7 - 2 = 5;\ M \cap C \text{ only} = 4 - 2 = 2;\ P \cap C \text{ only} = 3 - 2 = 1$
3. Step 3
  Single-set 'only' regions.
  $M \text{ only} = 18 - 5 - 2 - 2 = 9;\ P \text{ only} = 14 - 5 - 1 - 2 = 6;\ C \text{ only} = 10 - 2 - 1 - 2 = 5$
4. Step 4
  Total inside any set.
  $9 + 6 + 5 + 5 + 2 + 1 + 2 = 30$
5. Step 5
  Outside any set.
  $30 - 30 = 0$
Answer
$0$ students study none.
Examiner tip
Always work from the CENTRE outwards — never from the perimeter inwards.
3Shade a region
Extended• shading
Question
Shade the region representing $A' \cap B$ .
Step-by-step solution
1. Step 1
  $A'$ = everything outside $A$ . Intersect with $B$ .
2. Step 2
  Result: the part of $B$ that lies OUTSIDE $A$ .
Answer
Shade the part of $B$ that does not overlap $A$ .
4Find $n(A \cap B)$ from totals
Core• Adapted from 0580/22 May/Jun 2023 Q11• intersection, two set
Question
$30$ students play sport. $20$ play football (F), $15$ play tennis (T), and every student plays at least one of the two. Find $n(F \cap T)$ .
Step-by-step solution
1. Step 1
  Since every student plays at least one, $n(F \cup T) = 30$ .
2. Step 2
  Use inclusion-exclusion: $n(F \cup T) = n(F) + n(T) - n(F \cap T)$ .
  $30 = 20 + 15 - n(F \cap T)$
3. Step 3
  Rearrange.
  $n(F \cap T) = 35 - 30 = 5$
Answer
$5$ students play both.
5Probability from a two-set Venn diagram
Extended• probability, two set
Question
In a class of $25$ , $14$ have a brother (B), $10$ have a sister (S), $5$ have both. A student is chosen at random. Find (a) $P(B \cap S)$ , (b) $P(B' \cap S')$ (has neither).
Step-by-step solution
1. Step 1
  $B$ only $= 14 - 5 = 9$ ; $S$ only $= 10 - 5 = 5$ ; both $= 5$ .
2. Step 2
  Inside the circles: $9 + 5 + 5 = 19$ . Outside: $25 - 19 = 6$ .
3. Step 3
  Probabilities.
  $P(B \cap S) = \dfrac{5}{25} = \dfrac{1}{5};\ P(B' \cap S') = \dfrac{6}{25}$
Answer
(a) $\dfrac{1}{5}$ (b) $\dfrac{6}{25}$
6Fill in a three-set Venn diagram
Extended• Adapted from 0580/42 Feb/Mar 2024 Q9• three set, fill
Question
Of $50$ tourists: $28$ visited London (L), $24$ visited Paris (P), $20$ visited Rome (R). $12$ visited L and P, $10$ visited P and R, $8$ visited L and R. $5$ visited all three. Find the number who visited none.
Step-by-step solution
1. Step 1
  Centre.
  $L \cap P \cap R = 5$
2. Step 2
  Pairwise lens regions (subtract the centre).
  $L \cap P \text{ only} = 12 - 5 = 7;\ P \cap R \text{ only} = 10 - 5 = 5;\ L \cap R \text{ only} = 8 - 5 = 3$
3. Step 3
  Single-set 'only' regions.
  $L \text{ only} = 28 - 7 - 3 - 5 = 13;\ P \text{ only} = 24 - 7 - 5 - 5 = 7;\ R \text{ only} = 20 - 5 - 3 - 5 = 7$
4. Step 4
  Total inside the circles.
  $13 + 7 + 7 + 7 + 5 + 3 + 5 = 47$
5. Step 5
  Outside.
  $50 - 47 = 3$
Answer
$3$ tourists visited none.
7Find a union using inclusion-exclusion
Extended• union, two set
Question
$n(\mathcal{E}) = 60$ , $n(A) = 32$ , $n(B) = 28$ , $n(A \cap B) = 14$ . Find (a) $n(A \cup B)$ , (b) $n((A \cup B)')$ .
Step-by-step solution
1. Step 1
  Inclusion-exclusion.
  $n(A \cup B) = 32 + 28 - 14 = 46$
2. Step 2
  Complement.
  $n((A \cup B)') = 60 - 46 = 14$
Answer
(a) $46$ (b) $14$
8Word-problem Venn (two subjects)
Extended• Adapted from 0580/42 May/Jun 2024 Q12• word problem, two set
Question
In a year group of $80$ students, $45$ study Spanish, $33$ study French, and $12$ study neither language. Find how many students study both Spanish and French.
Step-by-step solution
1. Step 1
  Students studying at least one language $= 80 - 12 = 68$ .
2. Step 2
  Use $n(S \cup F) = n(S) + n(F) - n(S \cap F)$ .
  $68 = 45 + 33 - n(S \cap F)$
3. Step 3
  Rearrange.
  $n(S \cap F) = 78 - 68 = 10$
Answer
$10$ students study both.
Examiner tip
The examiner report flags that candidates often forget the 'neither' group and use $n(S \cup F) = 80$ directly, getting the wrong intersection.
9Three-set Venn with unknowns (Challenge)
Challenge• Adapted from 0580/42 Oct/Nov 2024 Q12• three set, algebra, challenge
Question
In a survey of $100$ people: $50$ liked apples (A), $40$ bananas (B), $35$ cherries (C). $20$ liked both A and B, $15$ liked both A and C, $10$ liked both B and C. Let $x$ = number who liked all three. Given that $5$ people liked none, find $x$ .
Step-by-step solution
1. Step 1
  Number liking at least one fruit $= 100 - 5 = 95$ .
2. Step 2
  Apply three-set inclusion-exclusion.
  $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)$
3. Step 3
  Substitute.
  $95 = 50 + 40 + 35 - 20 - 15 - 10 + x$
4. Step 4
  Simplify.
  $95 = 80 + x \Rightarrow x = 15$
Answer
$x = 15$ people liked all three.
Examiner tip
The examiner report consistently flags sign errors in the three-set formula. Remember: pairwise terms are SUBTRACTED, but the triple intersection is ADDED back in.

Key Formulae — Venn Diagrams

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

Two-set counting
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
When to use
Two overlapping sets.
Three-set counting
$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)$
When to use
Three overlapping sets.

Key Definitions and Keywords — Venn Diagrams

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

Region
Examiner keyword
A specific zone of a Venn diagram representing an exact combination of memberships.
$A$ only
Examiner keyword
Elements in $A$ but not in any other named set. = $A \setminus (B \cup C)$ in a 3-set diagram.
Shading a region
Examiner keyword
Marking the area corresponding to a named set expression, e.g. $A' \cap B$ .

Common Mistakes and Misconceptions — Venn Diagrams

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

✕Filling outer regions before the centre
0580/42 — every series
Why it happens
Reading questions left-to-right.
How to avoid it
Always start at $A \cap B \cap C$ and work outwards.
✕Putting the full pairwise count in the lens region
Why it happens
$n(M \cap P) = 7$ but the LENS region shows ONLY $M \cap P$ minus the centre.
How to avoid it
Pairwise lens region = $n(A \cap B) - n(A \cap B \cap C)$ .
✕Confusing ' $A$ only' with ' $A$ '
Why it happens
$A$ includes overlapping regions; ' $A$ only' excludes them.
How to avoid it
Read carefully — 'students who study Maths' includes overlaps; 'students who study ONLY Maths' excludes them.
✕Forgetting the universal-set region
Why it happens
Focusing on the named sets.
How to avoid it
After filling the circles, check that the rectangle adds up to $n(\mathcal{E})$ .

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Video lesson

Short walkthrough of the concepts students most often get stuck on.

Venn Diagrams — frequently asked questions

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

Venn Diagrams

Short Study Notes in the form of Slides

Short Study Notes — Venn Diagrams

Detailed Notes

What you’ll learn

How it’s examined

1Place elements in a two-set Venn diagram

2Counting with a three-set Venn diagram

3Shade a region

4Find n(A∩B)n(A \cap B)n(A∩B) from totals

5Probability from a two-set Venn diagram

6Fill in a three-set Venn diagram

7Find a union using inclusion-exclusion

8Word-problem Venn (two subjects)

9Three-set Venn with unknowns (Challenge)

Two-set counting

Three-set counting

Region

AAA only

Shading a region

✕Filling outer regions before the centre

✕Putting the full pairwise count in the lens region

✕Confusing 'AAA only' with 'AAA'

✕Forgetting the universal-set region

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Venn Diagrams — frequently asked questions

Venn Diagrams

Short Study Notes in the form of Slides

Short Study Notes — Venn Diagrams

Detailed Notes

What you’ll learn

How it’s examined

1Place elements in a two-set Venn diagram

2Counting with a three-set Venn diagram

3Shade a region

4Find n(A∩B)n(A \cap B)n(A∩B) from totals

5Probability from a two-set Venn diagram

6Fill in a three-set Venn diagram

7Find a union using inclusion-exclusion

8Word-problem Venn (two subjects)

9Three-set Venn with unknowns (Challenge)

Two-set counting

Three-set counting

Region

AAA only

Shading a region

✕Filling outer regions before the centre

✕Putting the full pairwise count in the lens region

✕Confusing 'AAA only' with 'AAA'

✕Forgetting the universal-set region

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Venn Diagrams — frequently asked questions

4Find $n(A \cap B)$ from totals

$A$ only

✕Confusing ' $A$ only' with ' $A$ '

4Find $n(A \cap B)$ from totals

$A$ only

✕Confusing ' $A$ only' with ' $A$ '