Sets

A set is a well-defined collection of objects, and almost every counting question in everyday life — how many students play football, how many families own a radio, how many people speak two languages — is really a question about sets. In Grade 10 you meet the one idea that makes these questions easy: a Venn diagram, filled in from the middle outwards. This chapter revises the language of sets, builds the two formulas for n(A\cup B) and n(A\cup B\cup C), and then solves the word problems that appear in the SEE paper almost every year.

1.1 Sets: concepts, formulas and worked problems

Language of sets — a quick revision

A set is a collection of well-defined objects. The objects are its elements (or members). We write x\in A for "x belongs to A" and x\notin A for "x does not belong to A".

Two ways to describe a set:

• Roster (listing) method: A=\{2,4,6,8\}.
• Set-builder (rule) method: A=\{x: x \text{ is an even natural number}, x<10\}.

Term	Meaning	Example
Finite set	has a countable number of elements	\{1,3,5\}
Empty set \emptyset	has no elements, n(\emptyset)=0	primes between 23 and 29
Universal set U	the set of everything we are talking about	U=\{1,2,3,\dots,10\}
Subset A\subseteq B	every element of A is also in B	\{2,4\}\subseteq\{2,4,6\}
Equal sets	same elements	\{1,2,3\}=\{3,1,2\}
Disjoint sets	no common element, A\cap B=\emptyset	\{1,2\} and \{5,6\}
n(A)	the number of elements of A	n(\{a,e,i\})=3

A set with n elements has 2^n subsets. For example \{a,b,c\} has 2^3=8 subsets.

The four operations

Two sets: the formula and the Venn diagram

When we add n(A) and n(B), the elements in A\cap B are counted twice. To count each element once we subtract the overlap one time.

Three sets

For three sets the same inclusion–exclusion idea gives a longer formula. Adding n(A)+n(B)+n(C) counts each pairwise overlap twice, so we subtract them; but that removes the triple overlap three times (after adding it three times), so we add it back once.

Set laws verified with given sets

To verify a law, evaluate the left-hand side and the right-hand side separately for the given sets and show they are the same set.

1.2 Exercises of the SEE type

Method for any survey question

1. Name the sets (T, C, …) and write each given number with its n(\;) symbol.
2. If "neither" is given, find n(A\cup B)=n(U)-n(\text{neither}).
3. Use the formula to find the overlap.
4. Draw the Venn diagram; fill the middle, then the "only" regions, then the outside.
5. Add all regions to check you get n(U).
6. Answer the specific question in a sentence.

Chapter summary

• n(A\cup B)=n(A)+n(B)-n(A\cap B); for disjoint sets n(A\cap B)=0.
• Only A: n(A)-n(A\cap B). Neither: n(U)-n(A\cup B).
• 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).
• Fill a three-set Venn diagram from the centre outwards; check that all regions add to n(U).
• Exactly two =\sum n(\text{pair})-3\,n(A\cap B\cap C); exactly one =n(A\cup B\cup C)-(\text{exactly two})-n(A\cap B\cap C).
• A-B=A\cap B' and A'=U-A.
• De Morgan: (A\cup B)'=A'\cap B', (A\cap B)'=A'\cup B'.
• Distributive: A\cap(B\cup C)=(A\cap B)\cup(A\cap C).

Practice questions

Group A — Multiple choice (1 mark each)

1. If n(A)=12, n(B)=9 and n(A\cap B)=4, then n(A\cup B) is [1]a25b17c21d13
2. In set notation, "elements of A that are not in B" is [1]aA\cup BbA\cap BcA\cap B'dA'\cap B
3. If U=\{1,2,3,\dots,10\} and A=\{2,3,5,7\}, then n(A') is [1]a4b5c6d10
4. If A\subseteq B, then A\cup B equals [1]aAbBc\emptysetdU
5. n(P\cup Q)=n(P)+n(Q) holds when [1]aP\subseteq QbP=QcP and Q are disjointdP'=Q

Answers to Group A

1. (b) — 12+9-4=17.

2. (c) — A-B=A\cap B'.

3. (c) — n(U)-n(A)=10-4=6.

4. (b) — every element of A is already in B.

5. (c) — nothing is counted twice only when P\cap Q=\emptyset.

Group B — Short answer (2 marks each)

1. If n(A)=22, n(B)=18 and n(A\cup B)=35, find n(A\cap B). [2]
2. Draw a Venn diagram and shade the region A'\cap B. [2]
3. If U=\{x: x\in\mathbb{N}, x\le 12\}, A=\{3,6,9,12\} and B=\{2,4,6,8,10,12\}, find A\cap B and A-B. [2]
4. In a class of 30 students every student plays at least one of football and volleyball. If 20 play football and 15 play volleyball, how many play both? [2]

Answers to Group B

1. n(A\cap B)=n(A)+n(B)-n(A\cup B)=22+18-35=5.

2. Draw U as a rectangle with overlapping circles A and B. A'\cap B is the part of B that lies outside A — the crescent of circle B not overlapped by A (the "only B" region).

3. A\cap B=\{6,12\} and A-B=\{3,9\}.

4. All 30 are in F\cup V: 30=20+15-n(F\cap V), so n(F\cap V)=35-30=5 students.

Group C — Short answer (3 marks each)

1. Of 100 students, 60 study Mathematics, 50 study Science and 20 study neither. Find how many study both, and how many study Mathematics only. [3]
2. In a group of 50 people, 30 drink milk, 25 drink tea and 10 drink neither. Find how many drink tea only. [3]
3. In a village, 70% of the families have a radio, 55% have a television and 20% have neither. Find the percentage that have both. If there are 400 families, how many have both? [3]
4. If A=\{1,2,3,4,5\}, B=\{2,4,6\} and C=\{3,4,7\}, verify A\cap(B\cup C)=(A\cap B)\cup(A\cap C). [3]

Answers to Group C

1. n(M\cup S)=100-20=80. n(M\cap S)=60+50-80=30. Mathematics only =60-30=30.

2. n(M\cup T)=50-10=40. n(M\cap T)=30+25-40=15. Tea only =25-15=10.

3. n(R\cup T)=100-20=80\%. Both =70+55-80=45\%. For 400 families: \frac{45}{100}\times 400=180 families.

4. B\cup C=\{2,3,4,6,7\}, so LHS =\{2,3,4\}. A\cap B=\{2,4\}, A\cap C=\{3,4\}, so RHS =\{2,3,4\}. LHS = RHS. ✓

Group D — Long answer (4 marks each)

1. In a survey of 100 students, 48 like English, 42 like Nepali and 36 like Mathematics. Also 18 like English and Nepali, 14 like Nepali and Mathematics, 12 like English and Mathematics, and 6 like all three. Draw a Venn diagram and find how many like(a) none of the subjects,(b) exactly one subject. [4]
2. In a group of 80 students, 12 like both tea and coffee and 8 like neither. The number who like tea only is three times the number who like coffee only. Find the number who like coffee. [4]
3. If U=\{1,2,3,\dots,12\}, A=\{1,3,5,7,9,11\} and B=\{2,3,5,7,11\}, verify De Morgan's law (A\cup B)'=A'\cap B'. [4]
4. Of 80 students, 40 play football, 35 volleyball and 30 basketball; 15 play football and volleyball, 12 volleyball and basketball, 10 football and basketball; 5 play all three. Find how many play exactly two games. [4]

Answers to Group D

1. Centre =6. Exactly two: E,N only =18-6=12; N,M only =14-6=8; E,M only =12-6=6. 1 (cont.) Exactly one: E only =48-12-6-6=24; N only =42-12-8-6=16; M only =36-8-6-6=16. Union =24+16+16+12+8+6+6=88 (formula: 48+42+36-18-14-12+6=88 ✓). (a) None =100-88=12. (b) Exactly one =24+16+16=56.

2. Let coffee only =y, tea only =3y. 3y+12+y+8=80, so 4y=60 and y=15. Students who like coffee =15+12=27.

3. A\cup B=\{1,2,3,5,7,9,11\}, so (A\cup B)'=\{4,6,8,10,12\}. A'=\{2,4,6,8,10,12\}, B'=\{1,4,6,8,9,10,12\}, so A'\cap B'=\{4,6,8,10,12\}. The two sides are equal. ✓

4. Exactly two =(15-5)+(12-5)+(10-5)=10+7+5=22 students.