Additional worked example 1 · Medium
Build a set from a rule
List \(A=\{\text{positive factors of }18\}\).
Step 1: Check which positive integers divide 18 without remainder.
Step 2: \(\boxed{A=\{1,2,3,6,9,18\}}\).
Chapter 2
Sets
Learn the language of sets and use Venn diagrams accurately. The chapter builds from basic notation to complements, subsets, two-set counting and exam-style three-set problems.
By the end of this chapter, you should be able to
- describe and list sets precisely using correct notation;
- use a universal set and the symbols \(\in\), \(\notin\), \(\cup\), \(\cap\), \(\varnothing\), \(A'\), \(\subseteq\), \(\not\subseteq\) and \(n(A)\);
- draw and interpret Venn diagrams, including disjoint and subset relationships;
- find unions, intersections and complements;
- work with subsets and the rule \(2^n\);
- solve two-set and three-set counting problems;
- read and write sets using list and set-builder notation.
1
Subtopic 1
The definition of a set
A set is a collection of objects that satisfy a clear condition. The objects are called elements or members of the set. Sets are normally written inside curly brackets.
When listing elements, put each distinct element only once and use braces. A description must be unambiguous: “integers from 2 through 6” gives \(\{2,3,4,5,6\}\); writing only “integers” would not specify this finite set.
List the elements
\(A=\{2,4,6,8\}\)
Describe the rule
\(A=\{\text{positive even integers less than }10\}\)
Important details
- A rule must be precise enough to decide whether an element belongs.
- Order does not matter: \(\{1,2,3\}=\{3,2,1\}\).
- Repeated entries are written once only: the letters in level form \(\{l,e,v\}\).
- A finite set has a fixed number of elements; an infinite set continues without end.
Worked example · Easy
Listing a set
List the elements of \(A=\{\text{factors of }20\}\).
Factors divide 20 exactly:
\[\boxed{A=\{1,2,4,5,10,20\}}.\]Additional worked example 2 · Exam-Style
Remove repeated elements
List the distinct letters in “STATISTICS” as a set and state the number of elements.
Step 1: Write each different letter once: S, T, A, I, C.
Step 2: \(A=\{S,T,A,I,C\}\) and \(\boxed{n(A)=5}\).
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
1.1
Easy [1 mark]
List the first five positive odd numbers as a set.
Show Answer
Step 1: The first five are 1, 3, 5, 7, 9.
Step 2: \(\boxed{\{1,3,5,7,9\}}\).
1.2
Medium [2 marks]
List the prime numbers smaller than 15.
Show Answer
Step 1: Prime numbers have exactly two factors.
Step 2: \(\boxed{\{2,3,5,7,11,13\}}\).
1.3
Exam-Style [3–4 marks]
List the distinct letters of “MATHEMATICS” and find \(n(A)\).
Show Answer
Step 1: Keep each letter once: M,A,T,H,E,I,C,S.
Step 2: \(\boxed{n(A)=8}\).
2
Subtopic 2
The universal set
The universal set, written \(\xi\), contains every element being considered in a particular problem. Smaller sets in the problem are taken from \(\xi\).
The universal set \(\xi\) fixes what is being considered. A complement depends on the chosen universal set, so it must be stated before identifying values outside a subset.
Membership
\(x\in A\) means “\(x\) is an element of \(A\)”.
\(x\notin A\) means “\(x\) is not an element of \(A\)”.
Why \(\xi\) matters
The same rule can give a different set when the universal set changes, because only elements inside \(\xi\) are allowed.
Worked example · Easy
Using the universal set
Let \(\xi=\{1,2,3,4,5,6,7,8,9,10\}\) and \(P=\{\text{prime numbers}\}\). List \(P\) and state whether \(9\in P\).
\[P=\{2,3,5,7\}.\]
Since 9 is not prime, \(\boxed{9\notin P}\).
Additional worked example 1 · Medium
Elements outside a subset
If \(\xi=\{1,2,3,4,5,6,7,8\}\) and \(A=\{2,4,6,8\}\), list \(A\) and all universal elements outside it.
Step 1: All of \(A\) is within \(\xi\).
Step 2: The outside elements are \(\boxed{\{1,3,5,7\}}\).
Additional worked example 2 · Exam-Style
Change the universal set
Let \(A=\{2,4,6\}\). Compare the elements outside \(A\) for \(\xi_1=\{1,2,3,4,5,6\}\) and \(\xi_2=\{1,2,\ldots,8\}\).
Step 1: For \(\xi_1\), outside \(A\) is \(\{1,3,5\}\).
Step 2: For \(\xi_2\), outside \(A\) is \(\{1,3,5,7,8\}\).
Step 3: The complement changes when the universal set changes.
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
2.1
Easy [1 mark]
If \(\xi=\{1,2,3,4,5\}\), can 6 belong to a subset of \(\xi\)?
Show Answer
Step 1: Every subset contains only elements of \(\xi\).
Step 2: \(\boxed{\text{No}}\), since 6 is not in the universal set.
2.2
Medium [2 marks]
Let \(\xi=\{1,2,\ldots,10\}\) and \(E\) contain all even elements. List the elements not in \(E\).
Show Answer
Step 1: \(E=\{2,4,6,8,10\}\).
Step 2: Not in \(E\): \(\boxed{\{1,3,5,7,9\}}\).
2.3
Exam-Style [3–4 marks]
Given \(\xi=\{2,3,4,5,6,7,8,9\}\), \(P\) is the set of primes in \(\xi\). Find \(n(P)\) and the number outside \(P\).
Show Answer
Step 1: Primes are \(\{2,3,5,7\}\), so \(n(P)=4\).
Step 2: There are eight universal elements, hence \(8-4=\boxed4\) outside.
3
Subtopic 3
Venn diagrams
A Venn diagram shows sets visually. The rectangle represents the universal set \(\xi\), while circles or ovals represent the sets inside it. Before filling a diagram, decide which region each element belongs to.
Begin with the overlap when two sets share members, then place the “only” members, then finish with anything outside both circles. Overlapping circles mean the sets may share elements; disjoint circles mean they share none.
PQ
Overlapping setsSome elements belong to both sets, so the circles overlap.
PQ
Disjoint setsNo element belongs to both sets, so \(P\cap Q=\varnothing\).
PQ
SubsetEvery element of \(Q\) also belongs to \(P\), so \(Q\subseteq P\).
A reliable way to fill a two-set Venn diagram
- Find the elements that belong to both sets and place them in the overlap first.
- Place the remaining elements that belong only to the first set.
- Place the remaining elements that belong only to the second set.
- Put any elements in \(\xi\) that belong to neither set outside both circles.
Venn-diagram rule
- Each element is written once only.
- The overlap means “in both sets”, not “write the element twice”.
- Always check the region outside the circles before finishing.
Worked example · Medium
Placing elements correctly
Let \(\xi=\{1,2,3,4,5,6,7,8,9,10,11,12\}\), \(P=\{\text{factors of }12\}\), and \(Q=\{\text{even numbers}\}\). Complete the Venn diagram.
First list each set:
\(P=\{1,2,3,4,6,12\}\)
\(Q=\{2,4,6,8,10,12\}\)
Both: \(\boxed{\{2,4,6,12\}}\)
\(P\) only: \(\boxed{\{1,3\}}\)
\(Q\) only: \(\boxed{\{8,10\}}\)
Neither: \(\boxed{\{5,7,9,11\}}\)
Visual check: overlap first, then the two “only” regions, then neither.
Additional worked example 1 · Medium
Fill a two-set Venn diagram
For \(\xi=\{1,2,3,4,5,6\}\), \(A=\{1,2,3\}\), \(B=\{3,4,5\}\), list each region.
Step 1: Both: \(A\cap B=\{3\}\).
Step 2: Only \(A=\{1,2\}\); only \(B=\{4,5\}\).
Step 3: Outside both \(=\{6\}\).
Additional worked example 2 · Exam-Style
Spot a nested set
Let \(A\) be multiples of 4 and \(B\) be even numbers, with \(\xi=\{1,\ldots,12\}\). Explain the Venn arrangement.
Step 1: \(A=\{4,8,12\}\) and \(B=\{2,4,6,8,10,12\}\).
Step 2: Every element of \(A\) is in \(B\), so the \(A\) circle lies entirely inside \(B\).
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
3.1
Easy [1 mark]
If \(A=\{1,2\}\), \(B=\{3,4\}\), what goes in the overlap?
Show Answer
Step 1: There are no common members.
Step 2: The overlap is \(\boxed{\varnothing}\).
3.2
Medium [2 marks]
Let \(A=\{2,4,6\}\), \(B=\{4,5,6\}\). Give the only-\(A\) and only-\(B\) regions.
Show Answer
Step 1: Common region is \(\{4,6\}\).
Step 2: Only \(A=\boxed{\{2\}}\); only \(B=\boxed{\{5\}}\).
3.3
Exam-Style [3–4 marks]
For \(\xi=\{1,\ldots,9\}\), \(A\) is even and \(B\) is a multiple of 3. List the four regions.
Show Answer
Step 1: \(A=\{2,4,6,8\}\) and \(B=\{3,6,9\}\).
Step 2: Both \(\{6\}\); only \(A\) \(\{2,4,8\}\); only \(B\) \(\{3,9\}\); outside \(\{1,5,7\}\).
Intersection means “in both”; union means “in at least one.” Count common elements only once in a union. Disjoint sets have intersection \(\varnothing\), but may still have a non-empty union.
4
Subtopic 4
The relationship between sets
Intersection
\(A\cap B\) contains elements in both \(A\) and \(B\).
Union
\(A\cup B\) contains elements in \(A\), in \(B\), or in both.
Empty set
\(\varnothing\) has no elements. If \(A\cap B=\varnothing\), the sets are disjoint.
Translate the notation into words
| Notation | Read it as | Region to think about |
|---|---|---|
| \(A\cap B\) | A and B | The overlap |
| \(A\cup B\) | A or B (or both) | Everything inside either circle |
| \(A\cap B'\) | In A and not in B | A-only region |
\(A\cap B\)
Shade only the overlap.
\(A\cup B\)
Shade both circles, including the overlap.
\(A\cap B'\)
Shade the part of A outside B.
Worked example · Easy
Union and intersection
Given \(A=\{2,4,6,8\}\) and \(B=\{3,6,9\}\), find \(A\cap B\) and \(A\cup B\).
The only element in both sets is 6:
\[A\cap B=\boxed{\{6\}}.\]The union contains every element appearing in either set, written once:
\[A\cup B=\boxed{\{2,3,4,6,8,9\}}.\]Worked example · Medium
Reading a region from notation
Describe \(A\cap B'\) in words.
\(B'\) means “not in B”. Intersecting this with A gives the elements that are in A but not in B.
So \(A\cap B'\) is the \(\boxed{\text{A-only region}}\).
Additional worked example 1 · Medium
Intersection and union
For \(A=\{1,3,5,7\}\) and \(B=\{3,6,7\}\), find \(A\cap B\) and \(A\cup B\).
Step 1: Common elements are 3 and 7, so \(\boxed{A\cap B=\{3,7\}}\).
Step 2: All distinct elements form \(\boxed{A\cup B=\{1,3,5,6,7\}}\).
Additional worked example 2 · Exam-Style
A union counting rule
In a club 25 play chess, 18 play draughts and 9 play both. Find how many play at least one.
Step 1: Add the group sizes: \(25+18=43\).
Step 2: Both were counted twice; subtract 9.
Step 3: \(n(C\cup D)=25+18-9=\boxed{34}\).
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
4.1
Easy [1 mark]
Find \(A\cap B\) for \(A=\{2,3,5\}\), \(B=\{3,4,5\}\).
Show Answer
Step 1: Find the elements in both sets.
Step 2: \(\boxed{\{3,5\}}\).
4.2
Medium [2 marks]
Find \(A\cup B\) for \(A=\{2,3,5\}\), \(B=\{3,4,5\}\).
Show Answer
Step 1: Combine distinct elements without repetition.
Step 2: \(\boxed{\{2,3,4,5\}}\).
4.3
Exam-Style [3–4 marks]
If \(n(A)=21\), \(n(B)=17\) and \(n(A\cup B)=30\), find \(n(A\cap B)\).
Show Answer
Step 1: Use \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\).
Step 2: \(30=21+17-x\), so \(\boxed{x=8}\).
5
Subtopic 5
The complement of a set
The complement \(A'\) contains every element of the universal set that is not in \(A\).
The complement is taken within \(\xi\), not among all possible numbers. For compound expressions, handle the brackets first: \((A\cup B)'\) means outside both, while \((A\cap B)'\) means anywhere except the shared region.
\[A'=\{x\in\xi:x\notin A\}.\]
\(A'\)
Everything in \(\xi\) except A.
\((A\cap B)'\)
Everything except the overlap.
\((A\cup B)'\)
Only the region outside both circles.
How to read brackets and complements
\((A\cap B)'\)
First find \(A\cap B\), then take everything not in that overlap.
\((A\cup B)'\)
First find \(A\cup B\), then take everything outside both sets.
The chapter also uses the equivalent relationships \((A\cap B)'=A'\cup B'\) and \((A\cup B)'=A'\cap B'\).
Worked example · Medium
Finding a complement
Let \(\xi=\{1,2,3,4,5,6,7,8,9,10\}\) and \(A=\{2,4,6,8,10\}\). Find \(A'\).
Remove the elements of \(A\) from the universal set:
\[\boxed{A'=\{1,3,5,7,9\}}.\]Worked example · Exam style
Complement of a union
Let \(\xi=\{1,2,3,\ldots,12\}\), \(A=\{2,4,6,8,10,12\}\), and \(B=\{3,6,9,12\}\). Find \((A\cup B)'\).
First combine everything in A or B:
\(A\cup B=\{2,3,4,6,8,9,10,12\}\).
Now choose the elements of \(\xi\) that are outside that union:
\[\boxed{(A\cup B)'=\{1,5,7,11\}}.\]Additional worked example 1 · Medium
Complement of a union
Let \(\xi=\{1,2,3,4,5,6,7\}\), \(A=\{1,2,3\}\), \(B=\{3,4\}\). Find \((A\cup B)^{\prime}\).
Step 1: \(A\cup B=\{1,2,3,4\}\).
Step 2: Within \(\xi\), the remaining elements are \(\boxed{\{5,6,7\}}\).
Additional worked example 2 · Exam-Style
Compare two complements
For the same sets, find \((A\cap B)^{\prime}\) and \(A^{\prime}\cup B^{\prime}\).
Step 1: \(A\cap B=\{3\}\), so its complement is \(\{1,2,4,5,6,7\}\).
Step 2: \(A^{\prime}=\{4,5,6,7\}\), \(B^{\prime}=\{1,2,5,6,7\}\).
Step 3: Their union is the same set, illustrating a De Morgan relationship.
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
5.1
Easy [1 mark]
In \(\xi=\{1,2,3,4,5\}\), let \(A=\{1,4\}\). Find \(A^{\prime}\).
Show Answer
Step 1: Remove the elements of \(A\) from \(\xi\).
Step 2: \(\boxed{\{2,3,5\}}\).
5.2
Medium [2 marks]
Let \(\xi=\{a,b,c,d,e\}\), \(A=\{a,b\}\), \(B=\{b,c\}\). Find \((A\cap B)^{\prime}\).
Show Answer
Step 1: \(A\cap B=\{b\}\).
Step 2: \(\boxed{\{a,c,d,e\}}\).
5.3
Exam-Style [3–4 marks]
In a class of 40, \(n(A\cup B)=33\). How many are in \((A\cup B)^{\prime}\)?
Show Answer
Step 1: The complement of a union means neither set.
Step 2: \(40-33=\boxed7\).
6
Subtopic 6
Subsets
\(A\subseteq B\) means every element of \(A\) is also an element of \(B\). A set is considered a subset of itself, and the empty set is a subset of every set.
A subset may equal the original set. The empty set is also a subset. A set of \(n\) distinct elements has \(2^n\) subsets; if the question asks for proper subsets, exclude the set itself.
Subset notation
\(A\subseteq B\): A is a subset of B.
\(A\not\subseteq B\): A is not a subset of B.
Number of subsets
If a set has \(n\) elements, the number of subsets is
\[\boxed{2^n}.\]Visual meaning of a subset
If \(Q\subseteq P\), every point in Q must lie inside P. Q may equal P, but it cannot contain an element outside P.
Example: if \(P=\{1,2,3,4\}\) and \(Q=\{1,3\}\), then \(Q\subseteq P\).
Worked example · Medium
Listing subsets
List all subsets of \(P=\{a,b,c\}\).
There should be \(2^3=8\) subsets:
\[\boxed{\varnothing,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}}.\]Common mistakes
- Do not forget \(\varnothing\) and the whole set when listing all subsets.
- Do not confuse \(x\in A\) (“x is an element”) with \(A\subseteq B\) (“A is a subset”).
Additional worked example 1 · Medium
Count all subsets
How many subsets does \(A=\{p,q,r,s,t\}\) have?
Step 1: There are \(n=5\) distinct elements.
Step 2: Number of subsets \(=2^5=\boxed{32}\).
Additional worked example 2 · Exam-Style
Proper subsets
A set has 4 elements. How many non-empty proper subsets can it have?
Step 1: Total subsets \(=2^4=16\).
Step 2: Exclude the empty set and the complete set.
Step 3: \(16-2=\boxed{14}\).
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
6.1
Easy [1 mark]
Is \(\{2,4\}\) a subset of \(\{1,2,3,4\}\)?
Show Answer
Step 1: Both 2 and 4 belong to the larger set.
Step 2: \(\boxed{\text{Yes}}\).
6.2
Medium [2 marks]
List every subset of \(\{a,b\}\).
Show Answer
Step 1: Include empty, each singleton and the set itself.
Step 2: \(\boxed{\varnothing,\{a\},\{b\},\{a,b\}}\).
6.3
Exam-Style [3–4 marks]
A set has 64 subsets. How many elements are in the set?
Show Answer
Step 1: Set \(2^n=64\).
Step 2: \(64=2^6\), so \(\boxed{n=6}\).
7
Subtopic 7
Problem solving with Venn diagrams
\(n(A)\) means the number of elements in set \(A\). In two-set problems, fill the intersection first so that people or objects counted in both sets are not counted twice.
For a two-set word problem, put the overlap in first, then calculate “only A” and “only B”. Check that four regions add to the full total: A only + both + B only + neither.
Two-set counting formula
\[\boxed{n(A\cup B)=n(A)+n(B)-n(A\cap B)}\]If the universal total is \(N\), then
\[\boxed{\text{neither}=N-n(A\cup B)}.\]Worked example · Exam style
Two-set survey
In a group of 50 students, 31 study French, 27 study Spanish and 15 study both. How many study neither language?
Put the overlap in first:
Both languages: \(15\).
French only: \(31-15=16\).
Spanish only: \(27-15=12\).
At least one language:
\[16+15+12=43.\]Neither:
\[50-43=\boxed{7}.\]Worked example · Medium
Finding an unknown intersection
For two sets \(A\) and \(B\), \(n(A)=28\), \(n(B)=35\), and \(n(A\cup B)=50\). Find \(n(A\cap B)\).
\[50=28+35-n(A\cap B)\]\[n(A\cap B)=63-50=\boxed{13}.\]
Additional worked example 1 · Medium
Find neither
Of 50 students, 28 study French, 23 study Spanish and 11 study both. How many study neither?
Step 1: At least one \(=28+23-11=40\).
Step 2: Neither \(=50-40=\boxed{10}\).
Additional worked example 2 · Exam-Style
Recover an overlap
In a group of 80, 46 play football, 39 play volleyball, and 12 play neither. How many play both?
Step 1: At least one \(=80-12=68\).
Step 2: \(68=46+39-x\).
Step 3: \(x=\boxed{17}\) people play both.
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
7.1
Easy [1 mark]
Of 24 people, 15 like tea and 8 like coffee, with 4 liking both. How many like at least one?
Show Answer
Step 1: Use \(15+8-4\).
Step 2: \(\boxed{19}\).
7.2
Medium [2 marks]
Of 35 students, 20 play basketball, 17 swim and 7 do both. Find the number doing neither.
Show Answer
Step 1: At least one \(=20+17-7=30\).
Step 2: Neither \(=35-30=\boxed5\).
7.3
Exam-Style [3–4 marks]
Of 70 visitors, 38 visited gallery A, 45 gallery B and 9 neither. How many visited only A?
Show Answer
Step 1: At least one \(=70-9=61\).
Step 2: Both \(=38+45-61=22\).
Step 3: Only A \(=38-22=\boxed{16}\).
8
Subtopic 8
Alternative ways to define a set
Finite sets can often be listed. Infinite or algebraically defined sets are more efficiently written using a rule. The colon in set-builder notation is read as “where” or “such that”.
A set may be defined by listing its elements or by a rule, such as \(\{x:x\in\mathbb N,\ 1\le x\le6\}\). A set of ordered pairs also needs a condition specifying which pairs are allowed.
| Form | Example | Meaning |
|---|---|---|
| List | \(\{1,4,9,16,25\}\) | The elements are written explicitly. |
| Rule | \(\{x:x\text{ is a natural number}\}\) | All natural-number values of \(x\). |
| Inequality | \(\{x:2<x<6\}\) | Values of \(x\) between 2 and 6. |
| Ordered pairs | \(\{(x,y):y=2x+1\}\) | All points satisfying the equation. |
Worked example · Medium
From rule to list
Let \(\xi=\{x:x\text{ is an integer}\}\) and \(P=\{x:-3<x\le2\}\). List \(P\).
The allowed integers are greater than −3 and up to 2 inclusive:
\[\boxed{P=\{-2,-1,0,1,2\}}.\]Additional worked example 1 · Medium
Convert a rule to a list
List \(B=\{x:x\text{ is an integer and }-2\le x<3\}\).
Step 1: Start at \(-2\) (included) and stop before 3.
Step 2: \(\boxed{B=\{-2,-1,0,1,2\}}\).
Additional worked example 2 · Exam-Style
Ordered-pair set
List \(T=\{(x,y):y=2x+1,\ x\in\{0,1,2,3\}\}\).
Step 1: For \(x=0,1,2,3\), calculate \(y=1,3,5,7\).
Step 2: \(\boxed{T=\{(0,1),(1,3),(2,5),(3,7)\}}\).
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
8.1
Easy [1 mark]
List \(\{x:x\text{ is a whole number, }0\le x<5\}\).
Show Answer
Step 1: Include 0, but exclude 5.
Step 2: \(\boxed{\{0,1,2,3,4\}}\).
8.2
Medium [2 marks]
Write \(\{4,9,16,25\}\) as a simple mathematical rule.
Show Answer
Step 1: These are squares of 2, 3, 4 and 5.
Step 2: \(\boxed{\{n^2:n\in\{2,3,4,5\}\}}\).
8.3
Exam-Style [3–4 marks]
List \(\{(x,y):y=x^2,\ x\in\{-2,-1,0,1\}\}\).
Show Answer
Step 1: Square each allowed value of \(x\).
Step 2: \(\boxed{\{(-2,4),(-1,1),(0,0),(1,1)\}}\).
9
Subtopic 9
Three-set problems
Three-set Venn diagrams have seven regions inside the circles plus the region outside all three. The key idea is to work from the most restricted region outward.
In a three-set diagram, fill the middle (all three) first. Subtract it from every stated pairwise intersection, then calculate each “only” section. Finally subtract all seven inside regions from the universal total.
Reliable order for a three-set problem
- Fill \(A\cap B\cap C\) first.
- Find the three pair-only regions by subtracting the triple intersection from each pair total.
- Find each single-only region by subtracting all known overlap regions from the set total.
- Add the seven inside regions.
- Subtract from the universal total to find “none”.
How to read three-set notation
\(A\cap(B\cup C)\)
Elements in A that are also in B or C. This includes the triple intersection.
\(A\cap B\cap C'\)
Elements in both A and B, but not in C: the A–B pair-only region.
Worked example · Exam style
Three-set survey
In a year group of 180 students, 95 study Mathematics \((M)\), 80 study Science \((S)\), and 70 study Art \((A)\). Also, 40 study \(M\cap S\), 32 study \(M\cap A\), 28 study \(S\cap A\), and 12 study all three. Find the number who study none.
Step 1: triple intersection = 12.
Step 2: pair-only regions
\[MS=40-12=28\]\[MA=32-12=20\]\[SA=28-12=16\]Step 3: single-only regions
\[M=95-28-20-12=35\]\[S=80-28-16-12=24\]\[A=70-20-16-12=22\]Step 4: total inside
\[35+24+22+28+20+16+12=157\]Step 5: none
\[180-157=\boxed{23}.\]Fill the centre first, then pair-only regions, then single-only regions.
Worked example · Exam style
Interpreting a three-set region
What does \(M\cap(T\cup P)\) represent?
First, \(T\cup P\) means “in T or P (or both)”. Intersecting with M keeps only the parts that are also in M.
So the required region is:
- \(M\cap T\cap P'\) (M and T only),
- \(M\cap P\cap T'\) (M and P only), and
- \(M\cap T\cap P\) (all three).
In words: \(\boxed{\text{students in M who also study T or P}}\).
Critical three-set warning
A statement such as \(n(A\cap B)=40\) normally includes those who are also in C. If 12 are in all three, the A–B only region is \(40-12=28\).
Additional worked example 1 · Medium
Exactly two sets
If \(n(A\cap B)=12\), \(n(A\cap B\cap C)=5\), how many are in \(A\cap B\) but not \(C\)?
Step 1: The pairwise intersection includes the three-way region.
Step 2: \(12-5=\boxed7\), which is \(n(A\cap B\cap C^{\prime})\).
Additional worked example 2 · Exam-Style
Three activities
In a group of 60, 25 do A, 22 do B, 19 do C; pairwise intersections are 10, 8 and 7; 4 do all three. Find how many do none.
Step 1: Use inclusion–exclusion for three sets.
Step 2: At least one \(=25+22+19-10-8-7+4=45\).
Step 3: None \(=60-45=\boxed{15}\).
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
9.1
Easy [1 mark]
If 9 belong to \(A\cap B\) and 3 to all three sets, how many belong to A and B only?
Show Answer
Step 1: Subtract the centre from the pairwise total.
Step 2: \(9-3=\boxed6\).
9.2
Medium [2 marks]
In 3 sets, 6 people belong to exactly one set, 9 to exactly two and 3 to all three. Find the number in the union.
Show Answer
Step 1: The inside regions are disjoint when grouped by how many sets.
Step 2: \(6+9+3=\boxed{18}\).
9.3
Exam-Style [3–4 marks]
Among 100 students, \(n(A)=40\), \(n(B)=35\), \(n(C)=30\), \(n(A\cap B)=12\), \(n(A\cap C)=10\), \(n(B\cap C)=8\), and \(n(A\cap B\cap C)=4\). Find the number doing none.
Show Answer
Step 1: At least one \(=40+35+30-12-10-8+4=79\).
Step 2: None \(=100-79=\boxed{21}\).
Chapter 2 practice
Start with notation and straightforward set work, then move into visual Venn-diagram questions and full exam-style problems. Try every question before opening Show Answer.
A. Build confidence
1Listing a set[2 marks]
List the set \(A=\{\text{multiples of 5 less than 30}\}\).
Show Answer
The positive multiples of 5 below 30 are 5, 10, 15, 20 and 25.
\[\boxed{A=\{5,10,15,20,25\}}.\]2Membership notation[3 marks]
Let \(P=\{2,3,5,7,11,13\}\). Write \(\in\) or \(\notin\) in each statement:
- \(7\;\square\;P\)
- \(9\;\square\;P\)
- \(13\;\square\;P\)
Show Answer
7 and 13 are listed in the set; 9 is not.
a) \(7\in P\)
b) \(9\notin P\)
c) \(13\in P\)
3Union and intersection[4 marks]
Given \(A=\{1,3,5,7,9\}\) and \(B=\{2,3,5,8\}\), find:
- \(A\cap B\)
- \(A\cup B\)
Show Answer
The elements common to both are 3 and 5:
\[A\cap B=\boxed{\{3,5\}}.\]The union contains every distinct element in either set:
\[A\cup B=\boxed{\{1,2,3,5,7,8,9\}}.\]4Complement[3 marks]
Let \(\xi=\{1,2,3,4,5,6,7,8,9,10,11,12\}\) and \(C=\{1,2,3,4,6,12\}\). Find \(C'\).
Show Answer
Remove the elements of C from the universal set:
\[\boxed{C'=\{5,7,8,9,10,11\}}.\]5Subsets[4 marks]
List all subsets of \(\{p,q\}\).
Show Answer
A set with 2 elements has \(2^2=4\) subsets:
\[\boxed{\varnothing,\{p\},\{q\},\{p,q\}}.\]6Subset or not?[3 marks]
Let \(A=\{1,2,3,4,5\}\), \(B=\{2,4\}\), and \(C=\{2,6\}\). Complete the statements using \(\subseteq\) or \(\not\subseteq\).
- \(B\;\square\;A\)
- \(C\;\square\;A\)
Show Answer
a) Both 2 and 4 are in A, so \(\boxed{B\subseteq A}\).
b) 6 is not in A, so \(\boxed{C\not\subseteq A}\).
B. Learn to read Venn diagrams
7Name the shaded region[2 marks]
The purple part of the diagram is shaded. Write the region in set notation.
Show Answer
The shading is inside A but outside B.
\[\boxed{A\cap B'}\]8Complement shading[2 marks]
The shaded region is outside both circles. Write it in set notation.
Show Answer
Everything inside A or B is excluded, so this is the complement of the union.
\[\boxed{(A\cup B)'}\]Equivalently, by De Morgan's law, \(A'\cap B'\).
9Read a completed diagram[5 marks]
The diagram shows the numbers of people in each region.
Find:
- \(n(A)\)
- \(n(B)\)
- \(n(A\cup B)\)
- the total number in the universal set.
Show Answer
a) A includes A-only and the overlap: \(18+7=\boxed{25}\).
b) B includes B-only and the overlap: \(11+7=\boxed{18}\).
c) Union means all three inside regions: \(18+7+11=\boxed{36}\).
d) Include the 4 outside both circles: \(36+4=\boxed{40}\).
10Number of subsets[2 marks]
A set has 7 elements. How many subsets does it have?
Show Answer
Use \(2^n\):
\[2^7=\boxed{128}.\]11Set-builder notation[3 marks]
Let \(P=\{x:x\text{ is an integer and }-2\le x<4\}\). List the elements of \(P\).
Show Answer
Start at −2 because it is included; stop before 4:
\[\boxed{P=\{-2,-1,0,1,2,3\}}.\]C. Develop your problem-solving
12Two-set counting[5 marks]
In a class of 40 students, 26 play football, 18 play basketball and 9 play both. Find the number who play neither.
Show Answer
At least one sport:
\[n(F\cup B)=26+18-9=35.\]Neither:
\[40-35=\boxed{5}.\]13Unknown intersection[5 marks]
For two sets \(A\) and \(B\), \(n(A)=28\), \(n(B)=35\), and \(n(A\cup B)=50\). Find \(n(A\cap B)\).
Show Answer
Use
\[n(A\cup B)=n(A)+n(B)-n(A\cap B).\]So
\[50=28+35-n(A\cap B)\]\[n(A\cap B)=63-50=\boxed{13}.\]14Find an unknown from a Venn diagram[5 marks]
In the diagram, \(n(A)=22\).
Find \(x\).
Show Answer
Set A contains the A-only region and the overlap:
\[2x+(x+1)=22\]\[3x+1=22\]\[3x=21\]\[\boxed{x=7}.\]D. Exam-style challenge
15Two-set survey[6 marks]
Of 75 people, 46 like tea, 38 like coffee and 21 like both. Find:
- the number who like tea only;
- the number who like coffee only;
- the number who like neither.
Show Answer
a) Tea only: \(46-21=\boxed{25}\).
b) Coffee only: \(38-21=\boxed{17}\).
c) At least one: \(25+21+17=63\). Therefore neither: \(75-63=\boxed{12}\).
16Three-set problem[8 marks]
In a group of 120 students, 62 study English \((E)\), 55 study History \((H)\), and 48 study Geography \((G)\). Also, 27 study \(E\cap H\), 24 study \(E\cap G\), 21 study \(H\cap G\), and 10 study all three. Find the number who study none.
Show Answer
First remove the triple intersection from the pair totals:
\[EH\text{ only}=27-10=17,\quad EG\text{ only}=24-10=14,\quad HG\text{ only}=21-10=11.\]Now single-only regions:
\[E\text{ only}=62-17-14-10=21\]\[H\text{ only}=55-17-11-10=17\]\[G\text{ only}=48-14-11-10=13.\]At least one:
\[21+17+13+17+14+11+10=103.\]None:
\[120-103=\boxed{17}.\]17Three-set notation[5 marks]
In a three-set Venn diagram, describe the region \(A\cap(B\cup C)\) in words and state which parts of A are included.
Show Answer
\(B\cup C\) means in B or C (or both). The intersection with A keeps only regions also inside A.
Included regions:
- A and B only,
- A and C only,
- all three sets.
In words: \(\boxed{\text{elements in A that are also in B or C}}\).
18Mixed set notation[6 marks]
Let \(\xi=\{1,2,3,\ldots,20\}\), \(A=\{\text{multiples of }2\}\) and \(B=\{\text{multiples of }3\}\). Find:
- \(A\cap B\)
- \((A\cup B)'\)
Show Answer
a) Numbers that are multiples of both 2 and 3 are multiples of 6:
\[A\cap B=\boxed{\{6,12,18\}}.\]b) First list the numbers in \(A\cup B\):
\(2,3,4,6,8,9,10,12,14,15,16,18,20\).
Numbers in \(\xi\) that are in neither set are
\[\boxed{(A\cup B)'=\{1,5,7,11,13,17,19\}}.\]Chapter 2 high-grade checklist
Set language
I can use \(\in,\notin,\cup,\cap,\varnothing,A',\subseteq,\not\subseteq\) and \(n(A)\) correctly.
Shaded regions
I can move between a Venn-diagram region and notation such as \(A\cap B'\), \((A\cup B)'\) and \(A\cap(B\cup C)\).
Counting
I fill the intersection first and can use \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\) to solve unknown and “neither” questions.
Three sets
I start with the triple intersection, subtract it from pair totals, then find single-only and outside regions.
Chapter 2 revision: This chapter follows the subtopic headings and core concepts in the supplied Cambridge O Level Mathematics, Second Edition. Figure crops come from the provided textbook. Additional questions are original exam-style practice, not reproduced Cambridge examination questions.