Sets and Venn diagrams
List elements of unions and intersections, apply the inclusion-exclusion principle to overlapping groups, and solve problems using two- and three-set Venn diagrams.
Worked examples
Finding the union and intersection of two sets
Straightforward
Problem
Sets and .
(a) List and state .
(b) List and state .
(a) List and state .
(b) List and state .
1
Identify by listing elements that appear in both sets.
2
Elements , and are shared and counted once only.
Find by listing all distinct elements from either set.
Elements , and are shared and counted once only.
Answer
(a) , . (b) , .
Applying the inclusion-exclusion principle
Moderate
Problem
In a year group of 80 students, 48 study Drama and 35 study Music. If 12 students study both subjects, how many students study neither?
1
State the inclusion-exclusion principle for two sets.
2
Seventy-one students study at least one of the two subjects.
Substitute the known values to find .
Seventy-one students study at least one of the two subjects.
3
Subtract from the total to find students who study neither.
Answer
9 students study neither Drama nor Music.
Solving a three-set Venn diagram problem
Challenging
Problem
In a group of 60 people, 32 like coffee, 28 like tea and 20 like juice. 14 like coffee and tea, 10 like coffee and juice, 8 like tea and juice, and 4 like all three beverages. Find:
(a) the number who like exactly one beverage
(b) the number who like none of the three beverages.
(a) the number who like exactly one beverage
(b) the number who like none of the three beverages.
1
Find the number who like only coffee by subtracting all overlaps and adding back those counted twice.
2
Apply the same formula for only tea and only juice.
3
Sum the three 'only' regions to answer part (a).
4
Apply the three-set inclusion-exclusion formula to find , then subtract from 60 for part (b).
Answer
(a) 28 people like exactly one beverage. (b) 8 people like none of the three beverages.
Practise
Q1·Straightforward
The symbol represents which of the following?
Explanation
The intersection contains every element that belongs to both and at the same time. The union contains elements in or or both.
Q2·Straightforward
Sets and . How many elements are in ?
Explanation
The elements that appear in both and are , and . Therefore and .
Q3·Straightforward
Sets and . How many elements are in ?
Explanation
Combining all distinct elements: . The elements and appear in both sets but are counted only once. Therefore .
Q4·Straightforward
A Venn diagram shows two overlapping circles inside a rectangle (the universal set). There are 7 elements only in , 4 elements only in , and 3 elements in both and . How many elements are in ?
Explanation
includes every element in or (or both). Adding the three regions: . The region 'in both' is counted once, even though it sits inside both circles.
Q5·Moderate
In a class of 30 students, 18 play a sport and 15 play a musical instrument. If 7 students do both, how many students play a sport or a musical instrument (or both)?
Explanation
Using the inclusion-exclusion principle:
The 7 students who do both would otherwise be counted twice, so they are subtracted once.
The 7 students who do both would otherwise be counted twice, so they are subtracted once.
Q6·Moderate
At a gym, 45 members do cardio and 30 members do weights. If 60 members do at least one of these activities, how many members do both?
Explanation
Using :
Fifteen members do both cardio and weights.
Fifteen members do both cardio and weights.
Q7·Moderate
A survey of 100 people finds that 55 own a smartphone and 40 own a tablet. If 25 people own both, how many own neither a smartphone nor a tablet?
Explanation
Using inclusion-exclusion:
These 70 people own at least one device. The number who own neither is:
These 70 people own at least one device. The number who own neither is:
Q8·Moderate
In a group of 50 students, 28 study French and 22 study Spanish. If 10 students study both languages, how many study neither?
Explanation
Using inclusion-exclusion:
Forty students study at least one language. The number who study neither is:
Forty students study at least one language. The number who study neither is:
Q9·Challenging
In a class of 40 students, 22 play tennis, 18 play cricket and 15 play basketball. 10 play tennis and cricket, 8 play tennis and basketball, 6 play cricket and basketball, and 3 play all three. How many students play exactly one sport?
Explanation
For each sport, subtract the overlaps then add back those counted twice:
Q10·Challenging
In a class of 40 students, 22 play tennis, 18 play cricket and 15 play basketball. 10 play tennis and cricket, 8 play tennis and basketball, 6 play cricket and basketball, and 3 play all three. How many students play exactly two sports?
Explanation
Students who play exactly two sports are those in a pairwise intersection but not all three:
Q11·Challenging
A club surveys 80 members about three activities: swimming, cycling and running. 40 swim, 35 cycle and 30 run. 18 swim and cycle, 15 swim and run, 12 cycle and run, and 6 do all three. How many members do exactly one activity?
Explanation
Calculating the 'only' region for each activity:
Q12·Challenging
A club surveys 80 members about three activities: swimming, cycling and running. 40 swim, 35 cycle and 30 run. 18 swim and cycle, 15 swim and run, 12 cycle and run, and 6 do all three. How many members do none of these three activities?
Explanation
Using the three-set inclusion-exclusion principle:
The number who do none of the three activities is:
The number who do none of the three activities is: