Sets: Grade 10 Math – Science Playground

Nepal’s 2021 census visited about 6.66 million households. It counted about 4.88 million with a mobile phone and about 4.86 million with a smartphone. Add the two and you get 9.74 million: more households than Nepal has. Households with both kinds were counted twice, once in each list.

So how many really have a phone? A news report said 26.8% have none: does that follow from the two numbers? By the end of section 1.1 you will be able to say, with a picture you met in Grade 9, the Venn diagram, and one idea: fill it from the middle outwards. The same idea answers every “tick all that apply” survey in this unit, and the sets question the SEE asks every year.

1.0 Review: sets and Venn diagrams

A set is a well-defined collection of things, called its elements or members. “Students of Grade 10 who like science” is a set: for any student, you can say whether they belong to it or not.

Here is what you need from Grade 9, in one place.

Idea	In words	In symbols
Listing method	write the elements inside curly brackets	A = {1, 3, 5}
Cardinal number	how many elements a set has	n(A) = 3
Universal set	the set of everything being talked about	U
Subset	every element of A is also in B	A ⊂ B
Union	elements in A or B or both	A ∪ B
Intersection	elements in both A and B	A ∩ B
Difference	elements in A but not in B (“A only”)	A − B
Complement	elements of U not in A	A

Two sets are overlapping when they have some elements in common, and disjoint when they have none. The number of elements of a set, n(A), is also called the cardinality of the set. This unit is all about cardinality: we rarely list the elements, we count them.

In a Venn diagram, the universal set is a rectangle and each set is a circle inside it. Two overlapping circles cut the rectangle into four regions, and every element sits in exactly one of them.

The roll-number survey

Students with roll numbers 1 to 15 in Grade 10 were asked whether they like Mathematics and whether they like Science. Each student’s roll number is written in the region that fits their answers, in Figure 1.1.

Figure 1.1 The roll-number survey. M: students who like Mathematics; S: students who like Science. Roll numbers 2, 8 and 15 like both; 9 and 14 like neither.

Notice something in the answers. n(M) + n(S) = 8 + 8 = 16, but only 5 + 3 + 5 = 13 students like at least one subject. The 3 students in the middle were counted in M and in S. That double count is the heart of this unit.

Drag a roll number into another region (or tap it, then tap where it should go). Watch the sets and their cardinal numbers change. Does the last line ever fail?

Drag a number to another region, or tap it and then tap where it goes.

Now tap the region where should go.

M =: {1, 2, 3, 4, 8, 11, 13, 15}; n(M) = 8
S =: {2, 5, 6, 7, 8, 10, 12, 15}; n(S) = 8
M ∩ S =: {2, 8, 15}; n(M ∩ S) = 3
neither =: {9, 14}; n(M ∪ S) = 2

n(M) + n(S) − n(M ∩ S) = 8 + 8 − 3 = 13, and n(M ∪ S) = 13 ✓

In Figure 1.1, which set is M ∩ S?

Pick an answer to see why.

1.1 Cardinality of two sets

Activity 1: coffee or tea

Let C be the set of students who like coffee and T the set who like tea. Fill the diagram in this order.

The middle first. 6 students like both, so write 6 where the circles overlap: n(C ∩ T) = 6.
Then each circle’s other part. The coffee circle holds 15 students in all, and 6 of them are already written in the middle. So the part of C outside T holds 15 − 6 = 9. In the same way, tea only is 10 − 6 = 4.
Then outside the circles. 5 like neither.
Add the four regions for the whole class: 9 + 6 + 4 + 5 = 24 students.

Figure 1.2 The coffee-and-tea survey, filled from the middle outwards: 6 like both, 9 coffee only, 4 tea only, 5 neither. The class has 24 students.

Why the middle first? Because “15 like coffee” includes the 6 who like both. If you wrote 15 in the coffee-only part, the 6 would be counted twice. Starting from the middle means every number you write is a part no other number overlaps.

Names for the four regions

Every two-set question is about these four regions, so learn their names.

Region	In words	In symbols
inside A only	A only	n₀(A) = n(A − B)
the overlap	both A and B	n(A ∩ B)
inside B only	B only	n₀(B) = n(B − A)
outside both	neither A nor B	n(A ∪ B)

The small zero in n₀(A) means “only”. Both ways of writing it are used in exams; write either.

The formulas, and why they work

If A and B are overlapping sets:

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$

When you add n(A) and n(B), the elements in the overlap are counted twice, once in each circle. Subtracting n(A ∩ B) once leaves every element counted exactly once. In the coffee survey: n(C ∪ T) = 15 + 10 − 6 = 19, which is 9 + 6 + 4.

The rest follow from the picture:

$\begin{aligned} n_0(A) &= n(A) - n(A \cap B) \\ n_0(B) &= n(B) - n(A \cap B) \\ n(A \cup B) &= n_0(A) + n_0(B) + n(A \cap B) \end{aligned}$

If A and B are disjoint, there is no overlap, so n(A ∪ B) = n(A) + n(B).
If every element of U is in A or B, then n(U) = n(A ∪ B).
If some elements are in neither, then n(U) = n(A ∪ B) + n(A ∪ B).

The words exam questions use

The question says	It means	Count
at least one (either, one or both)	A or B or both	n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
both	the overlap	n(A ∩ B)
exactly one, only one	one of them but not both	n₀(A) + n₀(B) = n(A) + n(B) − 2 n(A ∩ B)
at most one (not both)	everyone except the overlap	n(U) − n(A ∩ B)
neither, none	outside both circles	n(A ∪ B) = n(U) − n(A ∪ B)

Why −2 n(A ∩ B) for “exactly one”? Adding n(A) and n(B) counts the overlap twice, and “exactly one” wants it zero times, so it is taken away twice.

In a group, n(A) = 12 and n(B) = 8. What is the least possible value of n(A ∪ B)?

Pick an answer to see why.

In a class, 18 students play football, 14 play cricket and 5 play both. How many play exactly one of the two games?

Pick an answer to see why.

Practice the method on the screen first: the widget asks for one region at a time, in the right order.

Pick a survey. The tinted region is the one to fill next: type how many belong there and press Check.

In a class of 40 students, 22 like momo (Mo), 18 like chowmein (Ch), and 6 like both.

Of 60 people, 20 drink only tea (T), 15 drink only coffee (C), and 10 drink neither.

In a school, 70% like football (F), 50% like cricket (Cr) and 10% like neither. 120 students like both. How many students are there?

Step 1. Start in the middle. How many like both?

Step 2. 22 like momo. How many like momo only?

Step 3. 18 like chowmein. How many like chowmein only?

Step 4. The class has 40. How many like neither?

Done. 16 + 6 + 12 + 6 = 40 ✓ Only one dish: 16 + 12 = 28.

Step 1. An “only” number goes straight in. How many drink tea only?

Step 2. How many drink coffee only?

Step 3. How many drink neither?

Step 4. The four regions make 60. How many drink both?

Done. 20 + 15 + 15 + 10 = 60 ✓ Tea drinkers in all: 20 + 15 = 35.

Step 1. Work in percentages. What percentage like neither?

Step 2. At least one is 100% − 10% = 90%. What percentage like both?

Step 3. What percentage like football only?

Step 4. What percentage like cricket only?

Step 5. Both is 30%, and that is 120 students. How many students in all?

Done. 30% of 400 = 120 ✓ Football only: 40% of 400 = 160 students.

Not yet. The question says it: 6 like both.

Not yet. The 22 includes the 6 in the middle. Take them away.

Not yet. 18 − 6: the middle is already counted.

Not yet. Add the three regions inside the circles and take that from 40.

Not yet. “Only tea” is the part of T outside C: 20.

Not yet. “Only coffee” is given: 15.

Not yet. Outside both circles: 10.

Not yet. 60 − (20 + 15 + 10).

Not yet. Given: 10%.

Not yet. 70% + 50% counts the middle twice: 70 + 50 − 90.

Not yet. 70% − 30%.

Not yet. 50% − 30%.

Not yet. 1% is 120 ÷ 30 = 4 students. The whole is 100%.

Worked examples

In each one, read the question, try it yourself, then open the solution one step at a time. A full answer has three parts: say what each letter stands for, draw the Venn diagram, then calculate with a sentence at the end.

There is a quicker way to see part b: everything in percentages. At least one subject is 100% − 5% = 95%. Both is 75% + 55% − 95% = 35%. So 35% of the students are 21 students, and 1% is 21 ÷ 35 = 0.6: the total is 0.6 × 100 = 60.

Back to the census

What we can say for sure: at least 4.88 million households, nearly three in four, have a phone, and at most 1.78 million (26.8%) have none. The two numbers alone can’t say more. Here is why.

Let M be the households with a mobile phone and S the households with a smartphone, in millions: n(U) = 6.66, n(M) = 4.88, n(S) = 4.86. The formula says

$n(M \cup S) = 4.88 + 4.86 - n(M \cap S) = 9.74 - n(M \cap S)$

and the census table doesn’t give the overlap. But the overlap can’t be just anything:

The union can’t be bigger than all the households: 9.74 − n(M ∩ S) ≤ 6.66, so n(M ∩ S) ≥ 3.08. At least 3.08 million households were counted in both lists.
The overlap can’t be bigger than the smaller set: n(M ∩ S) ≤ 4.86. Then n(M ∪ S) = 4.88, every smartphone household is also in M, and S ⊂ M.

So between 4.88 and 6.66 million households have a phone. Slide the overlap and watch.

The census gives the two circles; the slider sets the overlap. Every position fits the census’s numbers. What happens to the households with no phone?

Households with both kinds, n(M ∩ S): 3.97 million

A phone of any kind, n(M ∪ S): 5.77 million
No phone at all: 0.89 million
Share with no phone: 13.4%

The overlap can’t be smaller than this: here every household has a phone, and 4.88 + 4.86 still counts 3.08 million of them twice.

Somewhere in between. Every value on the slider fits the census’s two numbers, so those two numbers alone can’t tell you how many households have no phone.

The overlap can’t be bigger than this: every smartphone household is also counted in M, so S ⊂ M and n(M ∪ S) = n(M).

So the news report’s 26.8% is right only if S ⊂ M: if the census’s “mobile phone” includes smartphones. If it means the ordinary kind only, a household with just a smartphone is outside M but still has a phone, and the share with none could be anything from 26.8% down to 0%. Which reading is right is a question of what the census asked, not of maths. That is the real lesson: before you add two counts, ask what is in each set and how they overlap.

Exercise 1.1

Question 1

Give examples of the cardinality of sets, and show them to your teacher.

Answer

For two sets A and B with A ⊂ B, find n(A ∪ B) and n(A ∩ B).

Answer

If A and B are overlapping sets, write the formula for n(A ∪ B).

Answer

The sets A and B have 12 and 8 elements. Find the least possible number of elements in A ∪ B.

Answer

Question 2

In Figure 1.8, 80 people are in set M, 90 are in set E and 15 are in neither. Find:

Figure 1.8 Exercise 1.1, question 2.

a.n₀(M)
b.n₀(E)
c.n(M)
d.n(E)
e.n(M ∪ E)
f.n(M ∩ E)
g.n(M ∪ E)
h.n(U)

Answer

Question 3

If n(U) = 200, n₀(M) = 2x, n₀(E) = 3x, n(M ∩ E) = 60 and n(M ∪ E) = 40, find x.

Answer

If n(U) = 350, n(A) = 200, n(B) = 220 and n(A ∩ B) = 120, find n(A ∪ B) and n(A ∪ B).

Answer

If n(A) = 35 and n(A) = 25, find n(U).

Answer

Of two sets P and Q, P has 40 elements, P ∪ Q has 60 and P ∩ Q has 10. How many elements has Q?

Answer

Question 4

In a survey of 180 students of a school, 45 like Nepali only and 60 like English only, but 15 like neither subject.

i.Show the information in a Venn diagram.
ii.Find the number of students who like both subjects.
iii.Find the number who like at least one subject.

Answer

In a survey of 1,200 students of a school, 100 like Mathematics only and 200 like Science only, but 700 like neither.

i.Show the information in a Venn diagram.
ii.Find the number who like both subjects.
iii.Find the number who like at least one subject.

Answer

In a survey of 60 students, 10 play football only and 20 play volleyball only, but 12 play neither game.

i.Show the information in a Venn diagram.
ii.Find the number who play both games.
iii.Find the number who play at least one game.

Answer

Question 5

A survey of 900 people of a community found that 525 read Madhupark, 450 read Yuba Manch, and 75 read neither magazine.

i.Show the information in a Venn diagram.
ii.Find the number who read both.
iii.Find the number who read only one.

Answer

In a survey of 150 people, 90 like modern songs and 70 like folk songs, but 30 like neither.

i.Show the information in a Venn diagram.
ii.Find the number who like both kinds of song.
iii.Find the number who like only modern songs.

Answer

In a survey of 360 players, 210 liked to play volleyball and 180 liked football, but 30 liked neither.

i.Show the information in a Venn diagram.
ii.Find the number who like both games.
iii.Find the number who like only one game.

Answer

Question 6

Of the students who sat an examination, 70% passed English and 60% passed Mathematics, 20% failed both, and 550 students passed both.

i.Show the information in a Venn diagram.
ii.Find the total number of students who sat the examination.
iii.Find how many passed English only.

Answer

In a survey of students who sat the Grade 10 examination, 60% wanted to study Science and 70% Management, while 10% wanted neither. 400 students wanted to study both.

i.Show the information in a Venn diagram.
ii.Find how many students took part in the survey.
iii.Find the number who wanted to study Science only.

Answer

In a survey of the people of a community, 65% ride a motorcycle and 35% ride a scooter, while 20% ride both. 200 people ride both.

i.Show the information in a Venn diagram.
ii.Find how many people took part in the survey.
iii.Find the number who ride motorcycles only.

Answer

Question 7

Among 95 people of a community, the numbers who drink tea and coffee are in the ratio 4 : 5. 10 people drink both, and 15 drink neither.

i.Show the information in a Venn diagram.
ii.Find the number who drink exactly one of tea or coffee.
iii.Find the number who drink at least one.

Answer

In a class of 64 students, each likes milk or curd or both. The numbers who like only milk and only curd are in the ratio 2 : 1, and 16 like both.

i.Show the information in a Venn diagram.
ii.Find the number who like milk.
iii.Find the number who like only one.

Answer

At a conference of 320 participants, 60 only sing and 100 only dance. The number who do neither is three times the number who do both.

i.Show the information in a Venn diagram.
ii.Find how many do neither.
iii.Find the number who do at most one of the two.

Answer

Questions 8 to 11

In a survey of 200 people of a community, the numbers who use only a laptop and only a mobile were in the ratio 2 : 3. 30% use both and 15% use neither.

i.Show the information in a Venn diagram.
ii.Find the number who use a laptop.
iii.Find how many use at most one of the two.

Answer

In a survey of 300 players, one-third play volleyball only. 60% of the rest play football only, and 60 play neither. Using a Venn diagram, find the ratio of the number who play volleyball to the number who play football.

Answer

10.

Of 65 players in a survey, 11 play volleyball only and 33 play cricket only. The number who play cricket is twice the number who play volleyball. Using a Venn diagram, find the number who play both and the number who play neither.

Answer

11.

In a survey of 80 people, 60 like only oranges and 10 like both oranges and apples. The number who like oranges is 5 times the number who like apples. Using a Venn diagram, find the number who like apples only and the number who like neither fruit.

Answer

1.2 Cardinality of three sets

With three sets, three circles overlap, and the rectangle is cut into eight regions: one for each set alone, one for each pair, one in the very middle for all three, and one outside all the circles.

You already know a three-set Venn diagram, though you may not have seen it drawn. Your blood type says which of three markers (antigens) sit on your red blood cells: A, B and RhD. Group A has the A marker, group B the B marker, AB both and O neither; “positive” means the RhD marker is there and “negative” that it isn’t. Three markers, each there or not: that is 2 × 2 × 2 = 8 blood types, one for each region.

Figure 1.25 The eight common blood types are the eight regions of a three-set Venn diagram. A, B and D are the people whose red cells carry the A, B and RhD markers. O− is outside all three circles: no markers at all.

Activity 2: two diagrams

Figure 1.26 Activity 2, diagram 1.

Figure 1.27 Activity 2, diagram 2: the same, except that 8 has moved outside, and 10 is there too.

Activity 3: filling three circles

The rule is the same as for two sets: the middle first, then outwards.

The very middle: 5 like all three.
The pairs: 15 like oranges and mangoes, but that 15 includes the 5 who like all three. So oranges and mangoes only is 15 − 5 = 10. In the same way, mangoes and bananas only is 20 − 5 = 15, and oranges and bananas only is 25 − 5 = 20.
Each fruit alone: the orange circle holds 40, and 5 + 10 + 20 = 35 of them are already written. Oranges only: 40 − 35 = 5. Mangoes only: 35 − (5 + 10 + 15) = 5. Bananas only: 50 − (5 + 15 + 20) = 10.
Outside: 30.
Add all eight regions: 5 + 5 + 10 + 10 + 15 + 20 + 5 + 30 = 100 students.

Figure 1.28 The fruit survey, filled from the middle outwards. O: oranges, M: mangoes, B: bananas.

The relations

If A, B and C are overlapping sets, the picture gives these relations. The “only” regions of each pair:

$\begin{aligned} n_0(A \cap B) &= n(A \cap B) - n(A \cap B \cap C) \\ n_0(B \cap C) &= n(B \cap C) - n(A \cap B \cap C) \\ n_0(A \cap C) &= n(A \cap C) - n(A \cap B \cap C) \end{aligned}$

Each whole circle is four regions:

$\begin{aligned} n(A) &= n_0(A) + n_0(A \cap B) + n_0(A \cap C) + n(A \cap B \cap C) \\ n(B) &= n_0(B) + n_0(A \cap B) + n_0(B \cap C) + n(A \cap B \cap C) \\ n(C) &= n_0(C) + n_0(A \cap C) + n_0(B \cap C) + n(A \cap B \cap C) \end{aligned}$

The union is the seven regions inside the circles, and the universal set adds the outside:

$\begin{aligned} n(A \cup B \cup C) &= n_0(A) + n_0(B) + n_0(C) + n_0(A \cap B) + n_0(B \cap C) + n_0(A \cap C) + n(A \cap B \cap C) \\ n(U) &= n(A \cup B \cup C) + n(\overline{A \cup B \cup C}) \end{aligned}$

A story from maths

Euler’s circles, Venn’s diagrams

Circles for groups are older than Venn. In the 1760s the Swiss mathematician Leonhard Euler taught logic to a German princess by letter, drawing circles inside and beside one another: “all A are B” as one circle inside another. But Euler only drew the overlaps that existed.

In 1880, John Venn, an English logician at Cambridge, had a better idea: draw every possible overlap first, as overlapping circles, and then shade or fill in what you know. A region with nothing in it still gets drawn. That is why our diagrams always show four regions for two sets and eight for three, even when one of them holds 0.

Venn also liked building machines. He built one that bowled cricket balls, and it is said to have bowled out one of the best Australian batsmen when Australia’s team visited Cambridge in 1909. Today a stained-glass window of a three-set Venn diagram remembers him at his college in Cambridge.

A tall arched stained-glass window: three overlapping circles in yellow, purple and blue, above the words John Venn, Fellow 1857–1923, President 1903–23 — **Figure 1.29** The Venn diagram window at Gonville and Caius College, Cambridge, where John Venn was a fellow for 66 years. Each overlap is a darker color, and the very middle, inside all three circles, is the darkest. Photo: Schutz; window designed by Maria McClafferty (1989), CC BY-SA 2.5

The formula for three sets

Most questions give the whole sets and the pairs, not the regions. For those there is one formula:

$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$

Why does it end with a plus? Count how many times the very middle is counted. Adding the three sets counts it 3 times. Taking away the three pairs takes it away 3 times, leaving 0. So it has to be added back once. Try it on the screen.

Press the three steps in order and watch how many times each region has been counted.

Running total (Example 1): 139
Really in the union: 113

48 + 51 + 40 = 139. Too many: each pair-only region is counted twice and the middle three times.

139 − 11 − 10 − 9 = 109. Too few: each pair contained the middle, so it has now gone completely (×0).

109 + 4 = 113. Every region is counted exactly once, so this is n(A ∪ B ∪ C).

The formula can also be proved from the two-set formula, by treating A ∪ B as one set:

$\begin{aligned} n(A \cup B \cup C) &= n\big((A \cup B) \cup C\big) \\ &= n(A \cup B) + n(C) - n\big((A \cup B) \cap C\big) \\ &= n(A) + n(B) - n(A \cap B) + n(C) - n\big((A \cap C) \cup (B \cap C)\big) \\ &= n(A) + n(B) - n(A \cap B) + n(C) - \big[n(A \cap C) + n(B \cap C) - n(A \cap B \cap C)\big] \\ &= n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C) \end{aligned}$

The second line uses the two-set formula for A ∪ B and C. The third uses it again, and the distributive law (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C). The fourth uses the two-set formula once more, since (A ∩ C) ∩ (B ∩ C) = A ∩ B ∩ C.

If the three sets are disjoint (no two overlap), every overlap is 0 and n(A ∪ B ∪ C) = n(A) + n(B) + n(C).

In a three-set Venn diagram, n(A ∩ B) = 12 and n(A ∩ B ∩ C) = 5. What is n₀(A ∩ B)?

Pick an answer to see why.

Worked examples

You can also solve Example 2 straight from the diagram: call the middle x, write every other region in terms of x (Science and Mathematics only is 26 − x, Science only is 86 − (26 − x) − x − (32 − x) = 28 + x, and so on), and add all eight regions to get 168 + x = 180. It is longer, but it is the method to use when the formula doesn’t fit the question.

Exercise 1.2

Question 1

Figure 1.32 shows the elements of the sets P, Q and R. Find:

Figure 1.32 Exercise 1.2, question 1.

a.n(P)
b.n(Q)
c.n(P ∪ Q ∪ R)
d.n₀(P)
e.n₀(R)
f.n(P ∩ R)
g.n(P ∪ Q ∪ R)
h.n₀(P ∩ Q)
i.n(P ∩ Q ∩ R)

Answer

Question 2

If U = {positive integers less than 30}, P = {multiples of 2 less than 30}, Q = {multiples of 3 less than 30} and R = {multiples of 5 less than 30}, show the sets in a Venn diagram and verify:

a.n(P ∪ Q) = n(P) + n(Q) − n(P ∩ Q)
b.n(P ∪ Q ∪ R) = n(P) + n(Q) + n(R) − n(P ∩ Q) − n(Q ∩ R) − n(R ∩ P) + n(P ∩ Q ∩ R)
c.n(P ∪ Q ∪ R) = n(P − Q) + n(Q − R) + n(R − P) + n(P ∩ Q ∩ R)

Answer

Question 3

If n(U) = 100, n(M) = 45, n(E) = 50, n(S) = 35, n(M ∩ E) = 20, n(E ∩ S) = 20, n(S ∩ M) = 15 and n(M ∩ E ∩ S) = 5, find n(M ∪ E ∪ S).

Answer

If n(U) = 105, n(A) = 40, n(B) = 35, n(C) = 30, n(A ∩ B) = 15, n(B ∩ C) = 12, n(A ∩ B ∩ C) = 6 and n(A ∪ B ∪ C) = 30, find n(A ∩ C).

Answer

If n(U) = 120, n(M) = 50, n(E) = 40, n(S) = 45, n(M ∩ E) = 15, n(E ∩ S) = 15, n(S ∩ M) = 15 and n(M ∪ E ∪ S) = 15, find n(M ∩ E ∩ S).

Answer

If n(A ∪ B ∪ C) = 105, n₀(A) = 25, n₀(B) = 25, n₀(C) = 15, n₀(A ∩ B) = 15, n₀(A ∩ C) = 10 and n(A ∩ B ∩ C) = 10, find n₀(B ∩ C).

Answer

Question 4

Of 90 students who sat an examination, 43 passed Science, 40 Mathematics and 38 Nepali. 13 passed Science and Mathematics, 18 Mathematics and Nepali, and 16 Science and Nepali; 6 passed all three.

i.Show the information in a Venn diagram.
ii.Find the number who passed none of the subjects.

Answer

In a survey of a group, 60 like tea, 45 coffee and 30 milk; 25 like coffee and tea, 20 milk and tea, 15 coffee and milk, and 10 like all three. Everyone surveyed likes at least one of them.

i.Show the information in a Venn diagram.
ii.Find how many people were surveyed.

Answer

In a survey of 60 students, 23 played volleyball, 15 basketball and 20 cricket. 7 played volleyball and basketball, 5 basketball and cricket, and 4 volleyball and cricket, while 15 played none of the games.

i.Show the information in a Venn diagram.
ii.Find how many played all three games.
iii.How many played volleyball and cricket only?

Answer

Question 5

Of the students who sat an examination, 40% passed Science, 45% Mathematics and 50% Nepali. 10% passed Science and Mathematics, 20% Mathematics and Nepali, and 15% Science and Nepali, while 5% failed all three.

i.Find the percentage who passed all three subjects.
ii.Find the percentage who passed only one subject.
iii.Find the percentage who passed exactly two subjects.
iv.Find the percentage who passed at least one subject.
v.Show the information in a Venn diagram.

Answer

Question 6

A survey asked some people of a community whether they read Yuba Manch, Madhupark or Muna. 30 read Yuba Manch and 25 Madhupark; 15 read both Yuba Manch and Muna, and 12 both Yuba Manch and Madhupark. 9 read Madhupark only, 11 Muna only, 5 Yuba Manch and Madhupark only, and 10 read none of them.

a.Show the information in a Venn diagram.
b.Find the total number of people surveyed.
c.Find the number who read exactly two magazines.
d.Find the number who read Muna.

Hint

Answer

Question 7

90 people were asked which films they like by language: 48 like Nepali films, 40 English and 31 Hindi; 24 like Nepali and English, 19 Hindi and English, and 6 like all three. 21 like none of them.

a.How many like both Nepali and Hindi films?
b.How many do not like Hindi films?
c.How many like neither Nepali nor Hindi films?

Answer

How the SEE asks it

In the SEE’s Compulsory Mathematics paper, Sets is always one question of 6 marks, with four parts at four levels (CDC’s specification grid):

Part	Level	Marks	What it asks
a.	knowledge	1	write something in cardinality notation, or a formula
b.	understanding	1	show the information in a Venn diagram
c.	application	3	calculate a region, the total, or an unknown
d.	higher ability	1	a twist: a ratio, a percentage, or “what if…”

The question is a survey of two or three sets, with Nepali places, subjects or games (CDC’s own model question has three). Here is one of each, with how they are marked.

Mixed exercise

Two overlapping sets A and B have n₀(A) = 16 + x, n₀(B) = 5x, n(A ∩ B) = y and n(A ∪ B) = x.

a.Show the information in a Venn diagram.
b.If n(A) = n(B), find n(A ∪ B).
c.If also n(U) = 50, find the ratio of n(A ∩ B) to n(A ∪ B).

Answer

A and B are subsets of the universal set U, with n(U) = 100, n(A − B) = 32 + x, n(B − A) = 5x, n(A ∩ B) = x and n(A ∪ B) = y.

a.Show the information in a Venn diagram.
b.If n(A) = n(B), find n(A ∩ B).
c.Find n(A ∪ B).
d.By what percentage is n(A ∪ B) more or less than n(A ∩ B)?

Answer

In a survey of 93 women of a community, 80 work in farming and 71 in sewing, while 10 do other work.

a.Find the cardinality of each region and show it in a Venn diagram.
b.Find how many women do both.
c.How many times the number who only farm is the number who only sew?

Answer

In a survey of 1,000 farmers in a community, 800 grew potatoes and 500 tomatoes, while 50 grew other crops only.

a.Find the cardinality of each region and show it in a Venn diagram.
b.Find the number who grow both.
c.Write the numbers who grow only potatoes and only tomatoes as a ratio.

Answer

In a survey of 400 people of a community, the numbers with only a motorcycle license and only a car license were in the ratio 5 : 3. One-fourth of the people had both licenses, and 60 had none.

a.Show the information in a Venn diagram.
b.How many people had each license?
c.Find the number who had only a motorcycle license.

Answer

In a school, 100 students like football, 80 volleyball and 120 cricket. 30 like football and volleyball, 20 volleyball and cricket, and 35 football and cricket. 10 like all three games, and 5 like none of them.

a.Write the information in cardinality notation.
b.Show it in a Venn diagram.
c.Find the total number of students in the school.
d.What percentage of the students like football only?

Answer

A survey of 45 people of a community: 25 speak Nepal Bhasa, 23 Tamang and 15 Maithili. 12 speak Nepal Bhasa and Tamang, 5 Nepal Bhasa and Maithili, and 10 Tamang and Maithili; 4 speak all three.

a.Show the information in a Venn diagram.
b.How many speak none of these three languages?
c.How many speak only one of them?
d.How many speak both Nepal Bhasa and Tamang but not Maithili?

Answer

Worksheet to print (PDF, 5 pages) Answer key for teachers

Harder worksheet (PDF, 4 pages) Answer key for teachers

Quiz yourself on this unit →

Practice every question in the book, and keep track of the ones you miss →

Where these facts come from

The census’s household and phone counts, and the 26.8% with no phone (the opening): the 2021 census, as tabulated by Nepali Telecom, “National Census 2078: internet and phone access in Nepal”, March 2023.
The minus sign in a Google search and the example jaguar speed -car (1.0): Google Search Help, “Refine Google searches”, read October 2026.
Google Photos’ face groups and the Filter that shows the photos with both people (1.1): Google Photos Help, “Set up & manage your face groups”, section “Filter a face group”, read October 2026.
Blood groups, antibodies, the universal donor and recipient, and cross-matching (1.2): Wikipedia, “Blood type”, sections on the ABO and Rh systems and on universal donors and recipients, read October 2026.

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