Pack A — Fluency — 11.1 Sets and Venn Diagrams

BronzeQuestions 1–10

1.
Let $U = \{1,2,3,4,5,6,7,8\}$ , $A = \{ {setA} \}$ and $B = \{ {setB} \}$ . List the elements of $A \cap B$ .
2.
Using $A = \{ {setA} \}$ and $B = \{ {setB} \}$ , list $A \cup B$ .
3.
$U = \{1,2,3,4,5,6,7,8,9,10\}$ and $A = \{ {setA} \}$ . List $A'$ .
4.
In a Venn diagram, $n(A) = 12$ , $n(B) = 9$ and $n(A \cap B) = 4$ . Find $n(A \cup B)$ .
5.
In a class of 30 students, 18 play tennis, 12 play hockey, and 5 play both. How many play neither?
6.
Let $A = \{ {setA} \}$ . State whether $5 \in A$ or $5 \notin A$ .
7.
Let $A = \{2, 4, 6\}$ and $B = \{ {setB} \}$ . State whether $A \subseteq B$ is true or false.
8.
List the elements of $\{x \in \mathbb{Z} : 1 \leq x \leq 5\}$ .
9.
Let $A = \{ {setA} \}$ and $B = \{ {setB} \}$ . Find $A \cap B$ .
10.
In a universal set $U$ with $n(U) = 50$ , $n(A) = 18$ . Find $n(A')$ .

SilverQuestions 11–20

11.
In a survey of 50 people, 30 like tea, 24 like coffee, and 8 like neither. How many like both?
12.
In a class of 30, 18 study French, 12 study Spanish and 7 study both. How many study only French?
13.
$U = \{1, 2, \ldots, 15\}$ , $A = \{$ multiples of 3 $\}$ , $B = \{$ factors of 12 $\}$ . Find $A \cap B$ .
14.
$n(A) = 22$ , $n(B) = 18$ , $n(C) = 15$ . Pair intersections: $n(A\cap B) = 8$ , $n(A\cap C) = 6$ , $n(B\cap C) = 5$ . Triple: $n(A\cap B\cap C) = 3$ . Find $n(A \cup B \cup C)$ .
15.
Write the set $\{x \in \mathbb{R} : -2 \leq x < 5\}$ in interval notation.
16.
$U = \{1, 2, 3, 4, 5\}$ , $A = \{1, 3, 5\}$ , $B = \{2, 3\}$ . List $(A \cup B)'$ .
17.
Are $A = \{$ letters in MATHS $\}$ and $B = \{$ letters in STAMH $\}$ equal sets?
18.
Given $n(U) = 50$ , $n(A) = 30$ , $n(B) = 20$ and $n(A \cup B)' = 8$ , find $n(A \cap B)$ .
19.
Two fair dice are rolled. State the size of the sample space and list the outcomes giving a sum of 5.
20.
Let $A = \{1,2,3,4\}$ and $B = \{3,4,5,6\}$ . List $(A \cup B) \setminus (A \cap B)$ .

GoldQuestions 21–30

21.
In a class of 40, 22 study French, 18 study Spanish, 15 study German. 8 study French \& Spanish, 6 study French \& German, 5 study Spanish \& German, and 3 study all three. How many study at least one language?
22.
For the same class (use the totals above), how many study exactly one of the three languages?
23.
$U = \{1, 2, \ldots, 15\}$ , $A = \{$ multiples of 3 $\}$ , $B = \{$ factors of 12 $\}$ . Find $P(A' \cap B)$ when an element is chosen at random.
24.
In a 2-set Venn diagram for $A$ and $B$ , describe in words the region $A \cap B'$ , and give the number-elements interpretation.
25.
In a survey of $n(U) = 100$ students, $n(A) = 60$ and $n(B) = 45$ . If $n(A \cap B) = x$ and $n(A \cup B)' = y$ , write an equation relating $x$ and $y$ .
26.
In a Venn diagram with three sets $A$ , $B$ , $C$ , the region " $A$ only" has 12 students, " $B$ only" has 9, " $C$ only" has 7. Each pair-only region has 4 students. The all-three region has 3 students. There are 5 students in none of the three sets. Find $n(U)$ .
27.
For $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$ , $A = \{1, 2, 5, 6\}$ , $B = \{2, 3, 6, 7\}$ , find $A' \cap B$ .
28.
A coin and a 6-sided die are tossed together. Write the sample space $S$ as a set, and give $n(S)$ .
29.
Use De Morgan's laws to rewrite $(A \cup B)'$ without a union.
30.
In a 2-set Venn diagram, " $A$ only" has $2x$ students, " $B$ only" has $x + 5$ , "both" has $x$ , "neither" has $4$ . If $n(U) = 25$ , find $x$ .

PlatinumQuestions 31–40

31.
Of 100 students, 55 like Maths, 48 like Science, 30 like Art; 22 like Maths \& Science, 15 like Maths \& Art, 10 like Science \& Art, and 6 like all three. How many like exactly one of the three subjects?
32.
Verify with $U = \{1,2,3,4,5,6\}$ , $A = \{1, 2, 3\}$ , $B = \{3, 4, 5\}$ that $(A \cup B)' = A' \cap B'$ .
33.
A class has 30 students. 18 study French, 16 study Spanish, and 4 study neither. Let $x$ = number studying both. Find $x$ , and the number studying exactly one of the two languages.
34.
A four-digit code is formed using the digits $1, 2, 3, 4, 5$ without repetition. (a) State the size of the sample space. (b) Let $E$ = "the code is even". Find $n(E)$ .
35.
In a class of 50 students, 30 take Maths (M), 20 take Physics (P), 25 take Chemistry (C). 10 take M and P, 8 take M and C, 5 take P and C, and 3 take all three. How many take **none** of the three?
36.
Express in set-builder notation, then in interval notation, the set of all real $x$ satisfying both $x \geq 2$ and $x < 7$ .
37.
Of 200 customers, 120 bought coffee, 80 bought a pastry, and 50 bought both. A customer is chosen at random from those who bought a pastry. (a) How many candidates are there? (b) Of those, how many also bought coffee?
38.
In a survey of 60 students, every student plays at least one of football (F), basketball (B), or tennis (T). 30 play F, 28 play B, 20 play T. 10 play F and B, 12 play F and T, 8 play B and T. If $x$ students play all three sports, find $x$ .
39.
List all subsets of $A = \{a, b, c\}$ . How many are there?
40.
A school newspaper survey says: "Of those who read the print edition, 60% also read the online edition." Let $P$ = "reads print" and $O$ = "reads online". Express the statement in set notation using $n(\cdot)$ , $\cap$ and $\cup$ .

Pack A — Fluency

Pack A · Fluency