S3 Mathematics Practice Book · Unit 1: Problems on sets

Unit 1 Problems on sets

Key unit competence: Use set notation, Venn diagrams and relations in solving related problems.

Key ideas

Venn diagrams represent sets as circles inside a rectangle for the universal set $U$ . Shade the region asked for: $A \cap B$ , $A \cup B$ , $A^{'}$ , $A - B$ , $(A \cup B)^{'}$ , etc.
For two sets: $n (A \cup B) = n (A) + n (B) - n (A \cap B)$ ; "only $A$ " $= n (A) - n (A \cap B)$ ; "neither" $= n (U) - n (A \cup B)$ .
For three sets: $n (A \cup B \cup C) = n (A) + n (B) + n (C) - n (A \cap B) - n (A \cap C) - n (B \cap C) + n (A \cap B \cap C)$ .
Strategy for three-set problems: start from the centre $n (A \cap B \cap C)$ , then fill the "exactly two" regions, then the "only one" regions, then the outside.
De Morgan's laws: $(A \cup B)^{'} = A^{'} \cap B^{'}$ and $(A \cap B)^{'} = A^{'} \cup B^{'}$ .
Words: "both" means intersection; "either ... or" (at least one) means union; "neither" means the complement of the union; "only" means one set and none of the others.

Formulas to know

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

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

(A \cup B)^{'} = A^{'} \cap B^{'}

(A \cap B)^{'} = A^{'} \cup B^{'}

Common misconceptions

" $n (A)$ means the students who like only $A$ ." No: $n (A)$ includes those who also like other sets.
"In three-set problems, the number in $A \cap B$ goes straight into one region." No: $A \cap B$ includes the centre $A \cap B \cap C$ ; subtract it first.

Worked examples

Example 1. In a class of 40, 22 play football (F), 18 volleyball (V), 15 basketball (B); 7 play F and V, 6 F and B, 5 V and B, 3 play all three. How many play none?

Solution

$n (F \cup V \cup B) = 22 + 18 + 15 - 7 - 6 - 5 + 3 = 40$ . So $40 - 40 = 0$ play none.

Example 2. $n (U) = 50$ , $n (A) = 30$ , $n (B) = 25$ , $n (A \cap B) = 12$ . Find $n (A^{'} \cap B^{'})$ .

Solution

$n (A \cup B) = 30 + 25 - 12 = 43$ . $n (A^{'} \cap B^{'}) = n ((A \cup B)^{'}) = 50 - 43 = 7$ .

Practice questions

Easier and harder questions are mixed, just like in a real exam.

Draw a Venn diagram and shade $A^{'} \cap B$ .
Show answer
Shade the part of circle $B$ that lies outside circle $A$ .
Common mistake: Shading everything outside $A$ , including the region outside both circles.
Explain why $n (A \cap B) \leq n (A)$ always.
Show answer
$A \cap B$ is a subset of $A$ , so it cannot have more elements than $A$ .
Common mistake: Thinking the intersection can be larger when $B$ is large.
In a school of 200 students, 120 study French, 90 study Kiswahili and 30 study neither. How many study both?
Show answer
At least one: 170. $120 + 90 - x = 170$ , $x = 40$ .
Common mistake: Ignoring the 30 who study neither.
50 students sat Maths and English. 35 passed Maths, 30 passed English and 2 failed both. How many passed both?
Show answer
Passed at least one: 48. $35 + 30 - x = 48$ , $x = 17$ .
Common mistake: Using 50 instead of 48.
Using De Morgan's laws, show that $(A \cup B)^{'}$ and $A^{'} \cap B^{'}$ are the same for $U = {1, \dots, 8}$ , $A = {1, 2, 3}$ , $B = {3, 4, 5}$ .
Show answer
$A \cup B = {1, \dots, 5}$ , so $(A \cup B)^{'} = {6, 7, 8}$ . $A^{'} = {4, \dots, 8}$ , $B^{'} = {1, 2, 6, 7, 8}$ , $A^{'} \cap B^{'} = {6, 7, 8}$ . Same.
Common mistake: Computing $A^{'} \cup B^{'}$ instead.
Express in set notation: (a) in $A$ and $B$ but not $C$ ; (b) in exactly one of $A$ and $B$ .
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(a) $(A \cap B) \cap C^{'}$ ; (b) $(A \cap B^{'}) \cup (A^{'} \cap B) = A Δ B$ .
Common mistake: Writing $A \cup B$ for (b).
Of 35 teachers, 20 drink tea, 17 drink coffee, 12 drink milk; 8 tea and coffee, 6 tea and milk, 5 coffee and milk, 2 all three. Find how many drink none.
Show answer
Union $= 20 + 17 + 12 - 8 - 6 - 5 + 2 = 32$ ; none $= 3$ .
Common mistake: Forgetting to add back the triple intersection.
For the same teachers, how many drink tea only?
Show answer
$20 - 8 - 6 + 2 = 8$ .
Common mistake: Subtracting the triple intersection twice without adding it back.

Matching and drag-and-drop

Matching

Match each item in column A with the correct answer in column B. Write the letter next to each number.

Column A	Column B
1. Which formula is correct for three sets?	a. $n (A \cup B \cup C) = \sum n (A) - \sum n (A \cap B) + n (A \cap B \cap C)$
2. $n (A) = 2 x$ , $n (B) = x + 5$ , $n (A \cap B) = 4$ and $n (A \cup B) = 25$ . Find $x$ .	b. 8
3. Using the same sets, find $A^{'} \cap B$ .	c. 15
4. $n (U) = 60$ and $n (A \cup B) = 45$ . How many are in neither set?	d. ${5, 6}$

Show answer

1 → a ( $n (A \cup B \cup C) = \sum n (A) - \sum n (A \cap B) + n (A \cap B \cap C)$ ) · 2 → b (8) · 3 → d ( ${5, 6}$ ) · 4 → c (15)

Add singles, subtract pairs, add back the triple.
Step 1. $2 x + (x + 5) - 4 = 25$ .
Step 2. $3 x + 1 = 25$ , so $x = 8$ .
Elements of $B$ not in $A$ .
$60 - 45 = 15$ .

Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
$B$ · 29 · $A \cap B$ · $n (A) + n (B)$
1. If $A$ and $B$ are disjoint, then $n (A \cup B)$ is .
2. If $A \subset B$ , then $A \cup B$ equals .
3. $n (A) = 20$ , $n (B) = 15$ , $n (A \cap B) = 6$ . Find $n (A \cup B)$ . →
Show answer
1: $n (A) + n (B)$ · 2: $B$ · 3: 29
- $n (A \cap B) = 0$ .
- All of $A$ is already in $B$ .
- $20 + 15 - 6 = 29$ .

Matching

Match each item in column A with the correct answer in column B. Write the letter next to each number.

Column A	Column B
1. With the same sets ( $n (M) = 20$ , $n (P) = 18$ , $n (C) = 15$ , $n (M \cap P) = 8$ , $n (M \cap C) = 6$ , $n (P \cap C) = 5$ , $n (M \cap P \cap C) = 3$ ), how many are in $P$ only?	a. 10
2. With the same sets, how many belong to exactly two of $M$ , $P$ , $C$ ?	b. 24
3. $n (M) = 20$ , $n (P) = 18$ , $n (C) = 15$ , $n (M \cap P) = 8$ , $n (M \cap C) = 6$ , $n (P \cap C) = 5$ and $n (M \cap P \cap C) = 3$ . Find $n (M \cup P \cup C)$ .	c. 37
4. 60 farmers grow at least one of maize, beans, potatoes. Each crop is grown by 30 farmers, 3 grow all three. How many grow exactly two crops?	d. 8

Show answer

1 → d (8) · 2 → a (10) · 3 → c (37) · 4 → b (24)

Step 1. $P$ only $= n (P) - n (M \cap P) - n (P \cap C) + n (M \cap P \cap C)$ .
Step 2. $18 - 8 - 5 + 3 = 8$ (the centre was subtracted twice, so add it back once).
Step 1. Remove the centre (3) from each pairwise intersection: $8 - 3 = 5$ , $6 - 3 = 3$ , $5 - 3 = 2$ .
Step 2. $5 + 3 + 2 = 10$ .
Step 1. $n (M \cup P \cup C) = 20 + 18 + 15 - 8 - 6 - 5 + 3$ .
Step 2. $= 53 - 19 + 3 = 37$ .
Step 1. Let $p$ = exactly two, $q$ = exactly one.
Step 2. $q + p + 3 = 60$ and $q + 2 p + 9 = 90$ .
Step 3. Subtract: $p + 6 = 30$ .
Step 4. $p = 24$ .

Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
4 · 14 · ${7, 8, 9, 10}$ · 6
1. $n (A) = 20$ and $n (A \cap B) = 6$ . How many are in $A$ only? →
2. $n (A) = 10$ , $n (B) = 12$ , $n (A \cup B) = 18$ . Find $n (A \cap B)$ . →
3. $U = {1, \dots, 10}$ , $A = {1, 2, 3, 4}$ , $B = {3, 4, 5, 6}$ . Find $(A \cup B)^{'}$ . →
Show answer
1: 14 · 2: 4 · 3: ${7, 8, 9, 10}$
- $20 - 6 = 14$ .
- $10 + 12 - 18 = 4$ .
- $A \cup B = {1, \dots, 6}$ ; complement ${7, 8, 9, 10}$ .

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