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Set Notation IGCSE Extended

Grade 10 · Pure Mathematics · Cambridge IGCSE 0580 · Age 14–16

Welcome to Set Notation!

Sets are one of the most fundamental ideas in all of mathematics. A set is simply a well-defined collection of objects. Set notation gives us a precise language to describe collections, relationships, and operations — from listing the prime numbers below 20 to describing all rational numbers. On the Cambridge IGCSE 0580 Extended paper, set questions appear every year, often combining Venn diagrams with algebra to find unknown values.

n(A∪B) = n(A) + n(B) − n(A∩B)  |  (A∪B)' = A'∩B'  |  (A∩B)' = A'∪B'

Learning Objectives

  • Use set notation: ∈, ∉, {}, ∅, ξ, n(A), set-builder notation
  • Find unions (A∪B), intersections (A∩B) and complements (A')
  • Understand and use subset notation: A⊂B and A⊆B
  • Draw and interpret Venn diagrams for 2 and 3 sets
  • Shade regions in Venn diagrams from set expressions
  • Use n(A∪B) = n(A) + n(B) − n(A∩B) to solve problems
  • Find unknown values from Venn diagrams using algebra
  • Classify numbers into natural, integer, rational and real sets

Set Symbols

∈, ∉, ∅, ξ, n(A), listing & set-builder

Operations

Union ∪, intersection ∩, complement '

Subsets

A⊂B proper, A⊆B improper

Venn Diagrams

2-set & 3-set, shading regions

Problem Solving

n formula, unknowns, algebra

Number Sets

ℕ, ℤ, ℚ, ℝ and relationships

Learn 1 — Set Notation & Symbols

What is a Set?

A set is a well-defined collection of distinct objects called elements or members. Sets are written using curly braces { }. The order of elements does not matter and each element appears only once.

Examples of sets:
A = {1, 2, 3, 4, 5} — the first five positive integers
B = {red, green, blue} — primary colours of light
C = {x : x is an even number, 1 ≤ x ≤ 10} = {2, 4, 6, 8, 10}

Core Symbols

SymbolMeaningExample
∈is an element of / belongs to3 ∈ {1, 2, 3, 4} is TRUE
∉is NOT an element of5 ∉ {1, 2, 3, 4} is TRUE
{ }set brackets — list elements inside{2, 4, 6, 8}
∅empty set — contains no elements∅ = {} — do NOT write {∅}
ξuniversal set — all elements under considerationξ = {1,2,3,...,10}
n(A)number of elements in set An({3,6,9}) = 3
n(∅)number of elements in the empty setn(∅) = 0

Listing Sets

To list a set, write all elements between curly braces, separated by commas. There are two common ways to define a set:

Roster / List form: Write every element explicitly.
A = {2, 3, 5, 7, 11} — the prime numbers less than 13

Set-builder notation: Describe the rule for membership.
B = {x : x is a prime number, x < 13}
Read as: "B is the set of all x such that x is a prime number less than 13."

The colon : means "such that". A vertical bar | is also used: {x | x > 3}
More set-builder examples:
{x : x > 3, x ∈ ℤ} = {4, 5, 6, 7, ...} — integers greater than 3
{x : 2 ≤ x ≤ 8, x ∈ ℕ} = {2, 3, 4, 5, 6, 7, 8}
{x : x² = 4} = {−2, 2}
When asked to list a set from set-builder notation, simply identify every element that satisfies the rule and write them out in curly braces. Always check the domain (what type of number x must be).

The Universal Set ξ and the Empty Set ∅

The universal set ξ (xi) contains every element being considered in a particular problem. All other sets in the problem are subsets of ξ. It is drawn as a rectangle in a Venn diagram.

The empty set ∅ has no elements at all. It is a subset of every set. Write ∅ or {} — never {∅} (that would be a set containing the empty set, which has one element).

Example: ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {even numbers} = {2, 4, 6, 8, 10}  →  n(A) = 5
B = {multiples of 3} = {3, 6, 9}  →  n(B) = 3
"Set of even multiples of 7 below 10" = ∅  →  n(∅) = 0
Common error: Writing {∅} for the empty set. The empty set is ∅ or {}, NOT {∅}. The set {∅} actually contains one element (the empty set itself), so n({∅}) = 1.

Set-Builder Notation in Depth

Set-builder notation {x : condition} is especially useful for infinite sets or sets defined by inequalities.

{x : x > 3} — all numbers greater than 3 (domain depends on context)
{x : x ∈ ℤ, −2 < x ≤ 5} = {−1, 0, 1, 2, 3, 4, 5}
{x : x is a factor of 24} = {1, 2, 3, 4, 6, 8, 12, 24}
{x² : x ∈ {1, 2, 3, 4}} = {1, 4, 9, 16}
In IGCSE exams, set-builder questions often ask you to either: (a) list all elements from a given rule, or (b) write a rule in set-builder notation for a given list. Practice both directions.

Learn 2 — Set Operations & Venn Diagrams

Union: A ∪ B

The union A ∪ B contains every element that is in A, or in B, or in both. Think of it as combining the two sets together (but each element is only listed once).

Example: A = {1, 2, 3, 4}   B = {3, 4, 5, 6}
A ∪ B = {1, 2, 3, 4, 5, 6}  →  n(A ∪ B) = 6
Note: 3 and 4 appear in both but are only written once.

Intersection: A ∩ B

The intersection A ∩ B contains only the elements that are in both A AND B simultaneously.

Example: A = {1, 2, 3, 4}   B = {3, 4, 5, 6}
A ∩ B = {3, 4}  →  n(A ∩ B) = 2

Disjoint sets: A = {1, 2} and B = {5, 6}
A ∩ B = ∅ — no common elements (circles don't overlap on Venn diagram)

Complement: A'

The complement A' (read "A prime" or "A complement") contains all elements in the universal set ξ that are NOT in A.

Example: ξ = {1, 2, 3, 4, 5, 6, 7, 8}   A = {2, 4, 6, 8}
A' = {1, 3, 5, 7} — everything in ξ but not in A
n(A) + n(A') = n(ξ) always
A useful check: n(A) + n(A') = n(ξ). If ξ has 10 elements and n(A) = 6, then n(A') = 4.

Subsets: A ⊂ B and A ⊆ B

A is a subset of B (written A ⊆ B) if every element of A is also in B. A is a proper subset of B (written A ⊂ B) if A ⊆ B and A ≠ B (B has at least one element not in A).

A = {2, 4}   B = {1, 2, 3, 4, 5}
A ⊆ B: TRUE (every element of A is in B)
A ⊂ B: TRUE (A ≠ B since B has extra elements)

C = {1, 2, 3}   D = {1, 2, 3}
C ⊆ D: TRUE    C ⊂ D: FALSE (they are equal sets)
Key facts about subsets:
• ∅ ⊆ A for every set A (the empty set is a subset of everything)
• A ⊆ A for every set A (every set is a subset of itself)
• If n(A) = k, then A has 2ᵏ subsets in total
Do not confuse ∈ (element of) with ⊂ (subset of). The number 3 ∈ {1,2,3} but {3} ⊂ {1,2,3}. An element is a single object; a subset is itself a set.

Venn Diagrams — 2 Sets

A Venn diagram uses circles inside a rectangle (ξ) to show relationships between sets. The overlapping region represents the intersection.

The four regions in a 2-set Venn diagram:
• A only (in A but not B): A ∩ B'
• B only (in B but not A): A' ∩ B
• Both A and B (overlap): A ∩ B
• Neither (outside both circles): A' ∩ B' = (A ∪ B)'
Shading A ∪ B: shade the entire interior of both circles (all three inner regions).
Shading A ∩ B: shade only the overlapping lens-shaped region.
Shading A': shade everything OUTSIDE circle A (including B-only and neither regions).
Shading A ∩ B': shade the part of A that does NOT overlap with B (left crescent only).

Venn Diagrams — 3 Sets

With three sets A, B, C there are 8 regions: the 7 regions inside the circles (including triple overlap) plus the "outside all" region.

The 8 regions (labelled by which sets contain each element):
1. A only: A ∩ B' ∩ C'
2. B only: A' ∩ B ∩ C'
3. C only: A' ∩ B' ∩ C
4. A and B only: A ∩ B ∩ C'
5. A and C only: A ∩ B' ∩ C
6. B and C only: A' ∩ B ∩ C
7. All three: A ∩ B ∩ C
8. None: A' ∩ B' ∩ C' = (A ∪ B ∪ C)'
When shading a 3-set region, identify it step by step: e.g., A ∩ B' means "in A but NOT in B" — shade the parts of A that don't overlap with B (regions 1 and 5 in the list above).

Learn 3 — Problem Solving with Sets

The Inclusion-Exclusion Formula

The key formula for counting elements in unions is:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

We subtract n(A ∩ B) because elements in both A and B are counted twice when we add n(A) + n(B).

Example: In a class of 30 students, 18 study French (F), 14 study Spanish (S), and 7 study both.
n(F ∪ S) = 18 + 14 − 7 = 25 students study at least one language.
Students who study neither = 30 − 25 = 5
Rearrangements of the formula:
n(A ∩ B) = n(A) + n(B) − n(A ∪ B)
n(A) = n(A ∪ B) − n(B) + n(A ∩ B)

Finding Unknowns in Venn Diagrams

Exam questions often place an unknown x in a Venn diagram region, then give you n(ξ) or some other total. Set up an equation and solve.

Example: ξ = 40 elements. The Venn diagram shows:
A only: 2x    A∩B: x+3    B only: 3x−1    Neither: 5
Since all regions must sum to 40:
2x + (x+3) + (3x−1) + 5 = 40
6x + 7 = 40  →  6x = 33  →  x = 5.5
Check: 11 + 8.5 + 16.5 + 5 = 41 — always verify!
Another type: You are told n(A) = 15, n(B) = 20, n(A∩B) = 8, n(ξ) = 35.
A only = n(A) − n(A∩B) = 15 − 8 = 7
B only = n(B) − n(A∩B) = 20 − 8 = 12
n(A∪B) = 7 + 8 + 12 = 27
Neither = 35 − 27 = 8
Strategy for Venn diagram problems: (1) Fill in the intersection first, (2) subtract to find the "only" regions, (3) use the total to find the "neither" region. Work from the inside outwards.

Number Sets

The standard number sets are nested inside each other, from smallest to largest:

ℕ — Natural numbers: {1, 2, 3, 4, ...} — positive integers (note: some definitions include 0)
ℤ — Integers: {..., −3, −2, −1, 0, 1, 2, 3, ...} — whole numbers, positive, negative and zero
ℚ — Rational numbers: numbers that can be written as p/q where p, q ∈ ℤ and q ≠ 0. Includes all terminating and recurring decimals. E.g. 0.75, −3/7, 0.333...
ℝ — Real numbers: all points on the number line, including irrationals. E.g. √2, π, e are in ℝ but not in ℚ.
The nesting: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ
Every natural number is an integer. Every integer is rational (e.g. 5 = 5/1). Every rational is real. But NOT every real is rational (√2, π are irrational).

Classification examples:
−4 ∈ ℤ, −4 ∈ ℚ, −4 ∈ ℝ but −4 ∉ ℕ
0.6̄ = 2/3 ∈ ℚ (recurring decimal = rational)
√5 ∈ ℝ only (irrational — cannot be written as p/q)
π ∈ ℝ only (transcendental — not even algebraic)
√4 = 2 ∈ ℕ (it simplifies to a whole number). But √3 ∉ ℚ (it cannot be expressed as a fraction). Always simplify surds before classifying.

De Morgan's Laws

De Morgan's Laws connect complements with unions and intersections:

(A ∪ B)' = A' ∩ B'     (A ∩ B)' = A' ∪ B'
In words:
"NOT (A or B)" = "NOT A and NOT B" — you must be outside both.
"NOT (A and B)" = "NOT A or NOT B" — you miss at least one.

Example: ξ = {1,...,10}, A = {2,4,6,8,10}, B = {1,2,3,4,5}
A ∪ B = {1,2,3,4,5,6,8,10}  →  (A∪B)' = {7,9}
A' = {1,3,5,7,9}   B' = {6,7,8,9,10}
A' ∩ B' = {7,9} ✓ — matches (A∪B)'
Use De Morgan's Laws when a question asks you to shade or describe the complement of a union or intersection. Instead of working it out directly, swap the operation and complement each set individually.

Example 1 — Basic Set Operations

Q: ξ = {1,2,3,4,5,6,7,8,9,10}, A = {2,4,6,8,10}, B = {1,2,3,4,5}.
Find (a) A∪B   (b) A∩B   (c) A'   (d) n(B')
(a) A∪B: All elements in A or B: {1,2,3,4,5,6,8,10} A1
(b) A∩B: Elements in both: {2,4} A1
(c) A': Elements in ξ but not A: {1,3,5,7,9} A1
(d) B' = {6,7,8,9,10}  →  n(B') = 5 A1

Example 2 — Using the Formula

Q: n(A) = 17, n(B) = 12, n(A∩B) = 5, n(ξ) = 30. Find (a) n(A∪B)   (b) n(A'∩B')
M1: n(A∪B) = n(A) + n(B) − n(A∩B) = 17 + 12 − 5 = 24 A1
M1: n(A'∩B') = n((A∪B)') = n(ξ) − n(A∪B) = 30 − 24 = 6 A1

Example 3 — Unknown in Venn Diagram

Q: In a Venn diagram, A only = x, A∩B = 4, B only = 2x−1, Neither = 3. Total n(ξ) = 30. Find x and n(A).
M1: All regions sum to total: x + 4 + (2x−1) + 3 = 30
M1: 3x + 6 = 30  →  3x = 24  →  x = 8 A1
A1: n(A) = (A only) + (A∩B) = 8 + 4 = 12 A1

Example 4 — Set-Builder Notation

Q: List the elements of A = {x : x² < 30, x ∈ ℕ}
M1: Need natural numbers where x² < 30.
1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36 (too big)
A1: A = {1, 2, 3, 4, 5} A1

Example 5 — Subsets

Q: P = {factors of 12}, Q = {factors of 36}. Show P ⊂ Q.
P = {1,2,3,4,6,12}   Q = {1,2,3,4,6,9,12,18,36}
M1: Every element of P (1,2,3,4,6,12) is also in Q.
A1: P ⊆ Q. Since Q contains 9, 18, 36 which are not in P, P ≠ Q, so P ⊂ Q (proper subset). A1

Example 6 — De Morgan's Law

Q: ξ = {integers 1–12}, A = {multiples of 2}, B = {multiples of 3}. Verify (A∩B)' = A'∪B'.
A = {2,4,6,8,10,12}, B = {3,6,9,12}
A∩B = {6,12}  →  (A∩B)' = {1,2,3,4,5,7,8,9,10,11}
A' = {1,3,5,7,9,11}, B' = {1,2,4,5,7,8,10,11}
A1: A'∪B' = {1,2,3,4,5,7,8,9,10,11} ✓ Matches (A∩B)' A1

Common Mistakes in Set Notation

These are the errors that cost students marks most often on IGCSE set notation questions.

Mistake 1 — Confusing ∈ (element of) with ⊂ (subset of)

✗ Wrong: {3} ∈ {1, 2, 3, 4}   or   3 ⊂ {1, 2, 3, 4}
✓ Correct: 3 ∈ {1, 2, 3, 4}   and   {3} ⊂ {1, 2, 3, 4}

∈ connects an element (single object) to a set. ⊂ connects a set to another set. The number 3 is an element; {3} is a set containing the number 3. These are different objects.

Mistake 2 — Forgetting to subtract the intersection when counting the union

✗ Wrong: n(A∪B) = n(A) + n(B) = 15 + 12 = 27 (when n(A∩B) = 4)
✓ Correct: n(A∪B) = 15 + 12 − 4 = 23

Elements in A∩B are counted once in n(A) AND once in n(B). Adding them gives a count of 2 for those elements, so you must subtract n(A∩B) once to correct it.

Mistake 3 — Shading the wrong region for the complement

✗ Wrong: Shading only the "B only" region for A'
✓ Correct: A' = everything OUTSIDE circle A — shade "B only" AND "neither" region

The complement A' includes everything in ξ that is not in A. In a 2-set Venn diagram, this means the B-only crescent AND the outside-both area. Students often forget the "neither" region.

Mistake 4 — Writing {∅} instead of ∅ for the empty set

✗ Wrong: The set with no elements is {∅}
✓ Correct: The empty set is ∅ or {} — n(∅) = 0

{∅} is a set that contains the empty set as its one element, so n({∅}) = 1. This is different from the empty set ∅ which has n(∅) = 0. This is a subtle but frequently penalised error.

Mistake 5 — Putting intersection values in the wrong regions of a Venn diagram

✗ Wrong: n(A) = 10, so writing 10 in the A circle (including the overlap)
✓ Correct: n(A) = 10, n(A∩B) = 3, so write 3 in the overlap and 10−3 = 7 in the A-only region

The number n(A) counts ALL elements in A, including those also in B. The value you write in the "A only" part of the diagram must exclude the intersection. Always subtract the intersection first.

Key Formulas — Set Notation

Formula / FactNotes
n(A∪B) = n(A) + n(B) − n(A∩B)Inclusion-exclusion — the most tested formula
n(A) + n(A') = n(ξ)A and its complement together make the whole universal set
(A∪B)' = A' ∩ B'De Morgan's First Law
(A∩B)' = A' ∪ B'De Morgan's Second Law
A ⊆ B ⟺ A∩B = AA is a subset of B iff their intersection is just A
A ⊆ B ⟺ A∪B = BEquivalent condition for subset
n(∅) = 0Empty set has no elements
∅ ⊆ A for all sets AEmpty set is a subset of every set
ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝThe nesting of standard number sets
A ∩ A' = ∅A set and its complement are disjoint
A ∪ A' = ξA set and its complement together form the universal set
Venn diagram: All region values must sum to n(ξ). Work inside-out: fill intersection first, then "only" regions, then "neither".

Quick Symbol Reference

∈ element of  |  ∉ not element of  |  ⊂ proper subset  |  ⊆ subset
∪ union (or)  |  ∩ intersection (and)  |  ' complement (not)
∅ empty set  |  ξ universal set  |  n(A) number of elements
ℕ naturals  |  ℤ integers  |  ℚ rationals  |  ℝ reals

Interactive Venn Diagram

Click a region to shade or unshade it. Use the buttons to shade standard set expressions. The expression box shows what the current shading represents.

Click a region or button to begin
How to read the diagram: The rectangle is ξ (universal set). Circle A is on the left, circle B on the right. The overlapping lens is A∩B. Pink shading shows the selected region.

Set Calculator

Enter the universal set and two sets A and B (comma-separated integers), then explore all set operations.

Enter sets and click Calculate.

Exercise 1 — Set Membership & Counting

1. A = {2, 4, 6, 8, 10, 12}. Find n(A).

2. ξ = {1,2,3,...,15}, A = {multiples of 3 up to 15}. Find n(A).

3. B = {x : x is a prime number, x < 20}. Find n(B).

4. ξ = {1,...,20}, A = {odd numbers}. Find n(A').

5. A = {factors of 36}. Find n(A).

6. ξ has 25 elements. n(A) = 11. Find n(A').

7. A = {x : x² = 49, x ∈ ℤ}. Find n(A).

8. B = {x : 3 < x < 10, x ∈ ℕ}. Find n(B).

Exercise 2 — Union, Intersection & Complement

ξ = {1,2,3,...,12}, A = {2,4,6,8,10,12}, B = {3,6,9,12} for all questions.

1. Find n(A∪B).

2. Find n(A∩B).

3. Find n(A').

4. Find n(B').

5. Find n(A∩B') — elements in A but not B.

6. Find n(A'∩B') — elements in neither A nor B.

7. Find n((A∪B)').

8. Find n(A'∪B).

Exercise 3 — Using n(A∪B) = n(A) + n(B) − n(A∩B)

1. n(A)=10, n(B)=8, n(A∩B)=3. Find n(A∪B).

2. n(A)=15, n(B)=12, n(A∩B)=5. Find n(A∪B).

3. n(A∪B)=20, n(A)=13, n(B)=11. Find n(A∩B).

4. n(A∪B)=25, n(A∩B)=4, n(A)=16. Find n(B).

5. n(ξ)=30, n(A∪B)=22. Find n((A∪B)').

6. n(A)=18, n(B)=14, n(A∩B)=6, n(ξ)=35. Find the number in neither A nor B.

7. n(A)=20, n(B)=20, A and B are disjoint. n(ξ)=50. Find n(A'∩B').

8. n(A∪B)=40, n(A)=n(B)=25. Find n(A∩B).

Exercise 4 — Finding Unknowns in Venn Diagrams

1. A only = x, A∩B = 3, B only = x+2, Neither = 4, n(ξ) = 25. Find x.

2. A only = 2x, A∩B = x, B only = 3x, Neither = 5, n(ξ) = 35. Find x.

3. Using Q4.1 values: once x is found, find n(A).

4. A only = 5, A∩B = x+1, B only = 2x, Neither = 4, n(ξ) = 30. Find x.

5. n(A) = 3x, n(B) = 2x, n(A∩B) = x, n(ξ) = 40, neither = 10. Find x.

6. Using Q4.5 values: find n(A∪B).

7. A only = x+4, A∩B = 2x, B only = x+1, Neither = 3, n(ξ) = 44. Find x.

8. Using Q4.7 values: find n(B).

Exercise 5 — Number Sets & Subsets

Answer 1 for TRUE and 0 for FALSE.

1. Is −5 ∈ ℤ? (1=yes, 0=no)

2. Is √2 ∈ ℚ? (1=yes, 0=no)

3. Is 0.75 ∈ ℚ? (1=yes, 0=no)

4. Is π ∈ ℝ? (1=yes, 0=no)

5. Is ℕ ⊂ ℤ? (1=yes, 0=no)

6. How many subsets does {a, b, c} have?

7. A = {2, 4}, B = {1, 2, 3, 4, 5}. Is A ⊂ B? (1=yes, 0=no)

8. How many subsets does {1, 2, 3, 4} have?

Practice — 25 Mixed Questions

🔵 = Non-calculator    🟢 = Calculator allowed

🔵 1. A = {letters in "MATHEMATICS"}. Find n(A). (Count distinct letters)

🔵 2. n(A)=12, n(B)=9, n(A∩B)=4. Find n(A∪B).

🔵 3. n(ξ)=50, n(A∪B)=38. Find n((A∪B)').

🔵 4. ξ={1,...,10}, A={2,4,6,8,10}. Find n(A').

🔵 5. A only=7, A∩B=5, B only=8, neither=3. Find n(ξ).

🔵 6. n(A∪B)=30, n(A)=18, n(A∩B)=6. Find n(B).

🔵 7. A = {x : x is a factor of 20}. Find n(A).

🔵 8. Is 0.333... ∈ ℚ? (1=yes, 0=no)

🔵 9. A only=x, A∩B=4, B only=2x, neither=2, n(ξ)=30. Find x.

🔵 10. n(A)=16, n(B)=16, n(A∩B)=6. Find n(A∪B).

🔵 11. ξ={1,...,20}, A={multiples of 4}. Find n(A').

🔵 12. How many subsets does {p, q} have?

🔵 13. A={3,6,9,12}, B={2,4,6,8,10,12}. Find n(A∩B).

🔵 14. From Q13, find n(A∪B).

🔵 15. n(A)=25, n(A')=15. Find n(ξ).

🔵 16. Is √9 ∈ ℕ? (1=yes, 0=no)

🔵 17. A only=9, A∩B=6, B only=5, n(ξ)=28. Find number in neither.

🔵 18. n(A∪B)=45, n(A)=28, n(B)=23. Find n(A∩B).

🔵 19. A={multiples of 2 up to 20}, B={multiples of 5 up to 20}. Find n(A∩B).

🔵 20. From Q19, find n(A∪B).

🔵 21. B = {x : 1 ≤ x ≤ 6, x ∈ ℕ}. Find n(B).

🔵 22. A only=2x+1, A∩B=x, B only=3x, neither=4, n(ξ)=40. Find x.

🔵 23. From Q22, find n(A).

🔵 24. A={2,4,6,8}, B={1,2,3,4}. Find n(A∩B').

🔵 25. n(ξ)=60, n(A)=35, n(B)=30, n(A∩B)=15. Find n(A'∩B').

Challenge — 12 Questions (IGCSE Extended Level)

1. n(A∪B∪C)=55, n(A)=25, n(B)=20, n(C)=22, n(A∩B)=8, n(A∩C)=6, n(B∩C)=7, n(A∩B∩C)=3. Find the number in exactly one set. (Use inclusion-exclusion for 3 sets)

2. In a class of 40: 25 like maths (M), 20 like science (S), x like both, 3 like neither. Find x.

3. A = {x : x is a prime, x < 30}. Find n(A).

4. Three overlapping sets A, B, C in ξ=50. A only=8, B only=6, C only=9, A∩B only=4, A∩C only=3, B∩C only=5, A∩B∩C=2, neither=?. Find neither.

5. A={x : x² − 5x + 6 = 0}. Find n(A). (Solve the quadratic)

6. n(A)=3k, n(B)=2k, n(A∩B)=k, n(ξ)=50, neither=14. Find k.

7. How many proper subsets does a set with 4 elements have? (A proper subset excludes the set itself)

8. A={multiples of 6 up to 60}, B={multiples of 4 up to 60}. Find n(A∪B).

9. Verify De Morgan: ξ={1..10}, A={1,3,5,7,9}, B={2,3,5,8}. Find n((A∪B)').

10. A only=x²−2, A∩B=x+3, B only=2x, neither=1, n(ξ)=30. Find the positive value of x.

11. In a group of 100 people: 60 speak English (E), 50 speak French (F), 30 speak both. Find n(E'∩F') — speak neither.

12. A = {x : |x| ≤ 3, x ∈ ℤ}. Find n(A).

Exam Style Questions

Mark-scheme style. Show all working in your book. Enter final answers here for self-marking.

Question 1 — Venn Diagram with Unknown [5 marks]

ξ = {students in a year group}, n(ξ) = 60.
A = {students who study Art}, B = {students who study Biology}.
n(A) = 25, n(B) = 30, n(A∩B) = x, n(A'∩B') = 10.

(a) Show that x = 5.
Enter n(A∪B) to verify:
(b) Find n(A∩B') — students who study Art but not Biology.
(c) A student is described as studying exactly one of Art or Biology. How many students is this?

Question 2 — Set Operations & Listing [4 marks]

ξ = {x : x is an integer, 1 ≤ x ≤ 15}
P = {x : x is a multiple of 3}    Q = {x : x is a multiple of 5}

(a) Find n(P∩Q).
(b) Find n(P∪Q).
(c) Find n(P'∩Q') — integers from 1 to 15 not in P or Q.

Question 3 — Three-Set Venn Diagram [6 marks]

In a survey of 80 people: 45 like football (F), 38 like cricket (C), 30 like tennis (T).
n(F∩C) = 18, n(F∩T) = 14, n(C∩T) = 12, n(F∩C∩T) = 6.

(a) Find n(F∪C∪T) using: n(F∪C∪T) = n(F)+n(C)+n(T)−n(F∩C)−n(F∩T)−n(C∩T)+n(F∩C∩T).
(b) Find the number who like none of the three sports.
(c) Find n(F∩C∩T') — those who like football and cricket but NOT tennis.

Question 4 — Number Classification [3 marks]

State whether each number belongs to ℕ, ℤ, ℚ, or ℝ only. Enter the size of the SMALLEST set it belongs to: 1=ℕ, 2=ℤ, 3=ℚ, 4=ℝ only.

(a) √16 — enter 1 (ℕ), 2 (ℤ), 3 (ℚ), or 4 (ℝ only)
(b) −7 — enter 1, 2, 3, or 4
(c) √7 — enter 1, 2, 3, or 4

Question 5 — De Morgan's Laws & Shading [4 marks]

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 2, 3, 4, 5}    B = {4, 5, 6, 7, 8}

(a) Find n((A∩B)').
(b) Verify De Morgan's: Find n(A'∪B') and confirm it equals your answer to (a).
(c) Find n(A∩B') — elements in A but not B.