Sets & Venn Diagrams in IGCSE Maths
Set notation, Venn diagrams with two and three sets, shading regions, and the probability questions that use them.
0580 · E1.2 / E8.3Set language & notation IGCSE specific
GCSE covers basic Venn diagrams, but IGCSE requires you to read and write formal set notation. You need to know every symbol below — they appear in questions on both Paper 2 and Paper 4.
\(\xi\) — the universal set (everything under consideration).
\(A \cup B\) — the union of \(A\) and \(B\). Elements in \(A\) or \(B\) or both.
\(A \cap B\) — the intersection of \(A\) and \(B\). Elements in \(A\) and \(B\).
\(A'\) — the complement of \(A\). Everything in \(\xi\) that is not in \(A\).
\(n(A)\) — the number of elements in set \(A\).
\(\in\) — "is an element of". So \(3 \in A\) means 3 belongs to set \(A\).
\(\notin\) — "is not an element of".
\(A \subseteq B\) — \(A\) is a subset of \(B\). Every element of \(A\) is also in \(B\).
\(\emptyset\) — the empty set, containing no elements.
IGCSE questions often define sets using set-builder notation. You need to be comfortable reading these:
\[ A = \{x : x \text{ is a prime number}\} \] \[ B = \{x : 1 \leqslant x \leqslant 10, \; x \in \mathbb{Z}\} \]The first defines \(A\) as the set of all prime numbers. The second defines \(B\) as the integers from 1 to 10 inclusive.
Two-set Venn diagrams
A two-set Venn diagram has a rectangle representing the universal set \(\xi\), with two overlapping circles for sets \(A\) and \(B\). This creates four distinct regions:
Every element in \(\xi\) goes into exactly one of these four regions. The overlap is \(A \cap B\). The region outside both circles represents elements in neither set.
\(\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\), \(A = \{2, 3, 5, 7\}\), \(B = \{1, 2, 3, 4\}\). Place each element in the correct region of a Venn diagram.
Start with the intersection. \(A \cap B = \{2, 3\}\) — these go in the overlap.
Then \(A\) only: \(\{5, 7\}\). And \(B\) only: \(\{1, 4\}\).
Everything else goes outside: \(\{6, 8, 9, 10\}\).
Shading regions IGCSE specific
IGCSE questions often ask you to shade a region described in set notation, or to write the set notation for a shaded region. The key is to break compound expressions into steps.
For example, \((A \cup B)'\) means: find \(A \cup B\) first, then take the complement — so shade everything outside both circles.
And \(A' \cap B\) means: find \(A'\) (everything not in \(A\)), then intersect with \(B\) — so shade the part of \(B\) that doesn't overlap with \(A\).
The region inside \(B\) but outside \(A\) is shaded. Write this in set notation.
The shaded region contains elements that are in \(B\) and not in \(A\). In set notation:
\[ A' \cap B \]You could also write \(B \cap A'\) — intersection is commutative, so the order doesn't matter.
Shade the region represented by \(A \cap B'\).
\(B'\) is everything outside \(B\). Intersecting with \(A\) gives the part of \(A\) that doesn't overlap with \(B\):
Three-set Venn diagrams Extended only
Extended candidates need to work with three overlapping circles, which creates eight distinct regions. The approach is the same as two sets, but you need to be more careful about where elements sit.
The key rule when filling in a three-set diagram: always start from the centre (the region where all three sets overlap, \(A \cap B \cap C\)) and work outwards. This prevents double-counting.
In a class of 30 students: 18 study French, 15 study Spanish, 10 study German, 8 study French and Spanish, 5 study French and German, 4 study Spanish and German, and 2 study all three. Find how many study none of these languages.
Start from the centre: \(A \cap B \cap C = 2\).
French and Spanish only (not German): \(8 - 2 = 6\).
French and German only (not Spanish): \(5 - 2 = 3\).
Spanish and German only (not French): \(4 - 2 = 2\).
French only: \(18 - 6 - 3 - 2 = 7\).
Spanish only: \(15 - 6 - 2 - 2 = 5\).
German only: \(10 - 3 - 2 - 2 = 3\).
Total inside the circles: \(7 + 5 + 3 + 6 + 3 + 2 + 2 = 28\).
None: \(30 - 28 = \boxed{2}\).
Centre first, then the two-set-only overlaps, then the single-set-only regions, then outside. If you go in any other order you will almost certainly double-count.
Sets & probability
Venn diagrams connect directly to probability on the IGCSE syllabus. If you know the number of elements in each region, finding a probability is straightforward:
\[ P(A) = \frac{n(A)}{n(\xi)} \]The notation \(P(A \cap B)\) and \(P(A \cup B)\) may be used in Venn diagram contexts. You should also be comfortable with the addition rule:
\[ \begin{aligned} n(A \cup B) &= n(A) + n(B) - n(A \cap B) \end{aligned} \]From Example 1: \(\xi = \{1, ..., 10\}\), \(A = \{2, 3, 5, 7\}\), \(B = \{1, 2, 3, 4\}\). A number is chosen at random.
(a) Find \(P(A \cap B)\).
(b) Find \(P(A \cup B)'\).
Part (a)
\(A \cap B = \{2, 3\}\), so \(n(A \cap B) = 2\).
\[ P(A \cap B) = \frac{2}{10} = \frac{1}{5} \]Part (b)
\(A \cup B = \{1, 2, 3, 4, 5, 7\}\), so \(n(A \cup B) = 6\) and \(n(A \cup B)' = 4\).
\[ P(A \cup B)' = \frac{4}{10} = \frac{2}{5} \]Exam-style question
The Venn diagram shows the number of elements in each region.
\(\xi = \{1, 2, 3, ..., 30\}\).
(a) Write down an expression, in terms of \(x\), for \(n(\xi)\). [1]
(b) Find the value of \(x\). [2]
(c) Find \(n(P \cap Q)\). [1]
(d) Find \(n(P \cup Q)'\). [1]
(e) A number is chosen at random from \(\xi\). Find the probability that it is in set \(P\) but not in set \(Q\). [1]
(f) Write the following in set notation: the set of elements that are in \(Q\) but not in \(P\). [2]
Part (a)
Add all four regions:
\[ \begin{aligned} n(\xi) &= (x + 4) + x + (2x - 1) + 3 \\[6pt] &= 4x + 6 \end{aligned} \]Part (b)
Since \(\xi = \{1, 2, ..., 30\}\), we know \(n(\xi) = 30\):
\[ \begin{aligned} 4x + 6 &= 30 \\[6pt] 4x &= 24 \\[6pt] x &= 6 \end{aligned} \]Part (c)
The intersection is the overlap region, which contains \(x\) elements:
\[ n(P \cap Q) = 6 \]Part (d)
\((P \cup Q)'\) is the region outside both circles:
\[ n(P \cup Q)' = 3 \]Part (e)
\(P\) but not \(Q\) is the "P only" region: \(x + 4 = 10\).
\[ \begin{aligned} P(\text{in } P \text{ not } Q) &= \frac{10}{30} \\[6pt] &= \frac{1}{3} \end{aligned} \]Part (f)
Elements in \(Q\) but not in \(P\):
\[ Q \cap P' \]This is a classic 8-mark Venn diagram question. It combines algebra (forming and solving an equation from a diagram), set notation (writing regions in formal notation), and probability (converting counts to fractions). Parts (a)–(d) are accessible, but part (f) is where students lose marks — practise converting region descriptions into notation.
Common mistakes
If 8 students study French and Spanish, that includes the 2 who study all three. The "French and Spanish only" region is \(8 - 2 = 6\), not 8.
\(A \cup B\) means "in \(A\) or \(B\) or both" — it's everything inside both circles. \(A \cap B\) is only the overlap. Mixing these up is the single most common error on this topic.
\((A \cup B)'\) is not the same as \(A' \cup B'\). Work out the expression inside the brackets first, then take the complement of the result.
There are always elements in \(\xi\) that don't belong to any named set. These go outside all circles. When the question says "find \(n(\xi)\)", add all regions including the outside.
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