Corrigé
11.1 Sets and Venn Diagrams
Pack A — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Let $U = \{1,2,3,4,5,6,7,8\}$, $A = \{ 1,2,3,4 \}$ and $B = \{ 3,4,5,6 \}$. List the elements of $A \cap B$. | $\{3, 4\}$ |
| 2 | Using $A = \{ 1,2,3 \}$ and $B = \{ 2,3,4,5 \}$, list $A \cup B$. | $\{1, 2, 3, 4, 5\}$ |
| 3 | $U = \{1,2,3,4,5,6,7,8,9,10\}$ and $A = \{ 2,4,6,8,10 \}$. List $A'$. | $\{1, 3, 5, 7, 9\}$ |
| 4 | In a Venn diagram, $n(A) = 12$, $n(B) = 9$ and $n(A \cap B) = 4$. Find $n(A \cup B)$. | 17 |
| 5 | In a class of 30 students, 18 play tennis, 12 play hockey, and 5 play both. How many play neither? | 5 |
| 6 | Let $A = \{ 2,4,6,8,10 \}$. State whether $ 5 \in A$ or $ 5 \notin A$. | $5 \notin A$ |
| 7 | Let $A = \{2, 4, 6\}$ and $B = \{ 2,4,6,8,10 \}$. State whether $A \subseteq B$ is true or false. | True |
| 8 | List the elements of $\{x \in \mathbb{Z} : 1 \leq x \leq 5\}$. | $\{1, 2, 3, 4, 5\}$ |
| 9 | Let $A = \{ 1,3,5 \}$ and $B = \{ 2,4,6 \}$. Find $A \cap B$. | $\emptyset$ (the empty set) |
| 10 | In a universal set $U$ with $n(U) = 50$, $n(A) = 18$. Find $n(A')$. | 32 |
| 11 | In a survey of 50 people, 30 like tea, 24 like coffee, and 8 like neither. How many like both? | 12 |
| 12 | In a class of 30, 18 study French, 12 study Spanish and 7 study both. How many study only French? | 11 |
| 13 | $U = \{1, 2, \ldots, 15\}$, $A = \{$multiples of 3$\}$, $B = \{$factors of 12$\}$. Find $A \cap B$. | $\{3, 6, 12\}$ |
| 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 $n(A \cup B \cup C)$. | 39 |
| 15 | Write the set $\{x \in \mathbb{R} : -2 \leq x < 5\}$ in interval notation. | $[-2, 5)$ |
| 16 | $U = \{1, 2, 3, 4, 5\}$, $A = \{1, 3, 5\}$, $B = \{2, 3\}$. List $(A \cup B)'$. | $\{4\}$ |
| 17 | Are $A = \{$letters in MATHS$\}$ and $B = \{$letters in STAMH$\}$ equal sets? | Yes |
| 18 | Given $n(U) = 50$, $n(A) = 30$, $n(B) = 20$ and $n(A \cup B)' = 8$, find $n(A \cap B)$. | 8 |
| 19 | Two fair dice are rolled. State the size of the sample space and list the outcomes giving a sum of 5. | Sample space has 36 outcomes. Sum 5: $\{(1,4),(2,3),(3,2),(4,1)\}$. |
| 20 | Let $A = \{1,2,3,4\}$ and $B = \{3,4,5,6\}$. List $(A \cup B) \setminus (A \cap B)$. | $\{1, 2, 5, 6\}$ |
| 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? | 39 |
| 22 | For the same class (use the totals above), how many study exactly one of the three languages? | 21 |
| 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. | $\dfrac{1}{5}$ |
| 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. | Elements that are in $A$ but **not** in $B$ (i.e. "$A$ only"). |
| 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 $y$. | $x - y = 5$ (equivalently $x = y + 5$) |
| 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 $n(U)$. | 48 |
| 27 | For $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$, $A = \{1, 2, 5, 6\}$, $B = \{2, 3, 6, 7\}$, find $A' \cap B$. | $\{3, 7\}$ |
| 28 | A coin and a 6-sided die are tossed together. Write the sample space $S$ as a set, and give $n(S)$. | $n(S) = 12$; e.g. $S = \{(H,1),(H,2),\ldots,(T,6)\}$ |
| 29 | Use De Morgan's laws to rewrite $(A \cup B)'$ without a union. | $A' \cap B'$ |
| 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 $x$. | $x = 4$ |
| 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? | 57 |
| 32 | Verify with $U = \{1,2,3,4,5,6\}$, $A = \{1, 2, 3\}$, $B = \{3, 4, 5\}$ that $(A \cup B)' = A' \cap B'$. | Both equal $\{6\}$. |
| 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. | $x = 8$; exactly one $= 18$ |
| 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 $n(E)$. | (a) 120; (b) 48 |
| 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? | 5 students |
| 36 | Express in set-builder notation, then in interval notation, the set of all real $x$ satisfying both $x \geq 2$ and $x < 7$. | $\{x \in \mathbb{R} : 2 \leq x < 7\} = [2, 7)$ |
| 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? | (a) 80; (b) 50 |
| 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 $x$. | $x = 2$ |
| 39 | List all subsets of $A = \{a, b, c\}$. How many are there? | 8 subsets: $\emptyset, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\}$ |
| 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 $\cup$. | $\dfrac{n(P \cap O)}{n(P)} = 0.6$ |
Pack B — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Let $U = \{1,2,3,4,5,6,7,8\}$, $A = \{ 2,4,6,8 \}$ and $B = \{ 1,2,3,4 \}$. List the elements of $A \cap B$. | $\{2, 4\}$ |
| 2 | Using $A = \{ 2,4,6 \}$ and $B = \{ 1,2,3,4 \}$, list $A \cup B$. | $\{1, 2, 3, 4, 6\}$ |
| 3 | $U = \{1,2,3,4,5,6,7,8,9,10\}$ and $A = \{ 1,3,5,7,9 \}$. List $A'$. | $\{2, 4, 6, 8, 10\}$ |
| 4 | In a Venn diagram, $n(A) = 15$, $n(B) = 10$ and $n(A \cap B) = 6$. Find $n(A \cup B)$. | 19 |
| 5 | In a class of 25 students, 14 play tennis, 10 play hockey, and 4 play both. How many play neither? | 5 |
| 6 | Let $A = \{ 1,3,5,7,9 \}$. State whether $ 5 \in A$ or $ 5 \notin A$. | $5 \in A$ |
| 7 | Let $A = \{2, 4, 6\}$ and $B = \{ 2,4,8,10 \}$. State whether $A \subseteq B$ is true or false. | False |
| 8 | List the elements of $\{x \in \mathbb{Z} : -2 \leq x < 3\}$. | $\{-2, -1, 0, 1, 2\}$ |
| 9 | Let $A = \{ 10,20,30 \}$ and $B = \{ 5,15,25 \}$. Find $A \cap B$. | $\emptyset$ |
| 10 | In a universal set $U$ with $n(U) = 40$, $n(A) = 15$. Find $n(A')$. | 25 |
| 11 | In a survey of 40 people, 25 like tea, 20 like coffee, and 5 like neither. How many like both? | 10 |
| 12 | In a class of 40, 22 study French, 15 study Spanish and 6 study both. How many study only French? | 16 |
| 13 | $U = \{1, 2, \ldots, 15\}$, $A = \{$multiples of 3$\}$, $B = \{$factors of 12$\}$. Find $A \cap B$. | $\{1, 2, 3, 4, 6, 9, 12, 15\}$ |
| 14 | $n(A) = 30$, $n(B) = 24$, $n(C) = 20$. Pair intersections: $n(A\cap B) = 10$, $n(A\cap C) = 8$, $n(B\cap C) = 7$. Triple: $n(A\cap B\cap C) = 4$. Find $n(A \cup B \cup C)$. | 53 |
| 15 | Write the set $\{x \in \mathbb{R} : x > 3\}$ in interval notation. | $(3, \infty)$ |
| 16 | $U = \{1, 2, 3, 4, 5\}$, $A = \{1, 3, 5\}$, $B = \{2, 3\}$. List $(A \cup B)'$. | $\{1, 2, 4, 5\}$ |
| 17 | Are $A = \{$letters in MATHS$\}$ and $B = \{$letters in STAMH$\}$ equal sets? | Yes (repeats do not count in a set) |
| 18 | Given $n(U) = 50$, $n(A) = 30$, $n(B) = 20$ and $n(A \cup B)' = 5$, find $n(A \cap B)$. | 5 |
| 19 | Two fair dice are rolled. State the size of the sample space and list the outcomes giving a sum of 9. | Sample space has 36 outcomes. Sum 9: $\{(3,6),(4,5),(5,4),(6,3)\}$. |
| 20 | Let $A = \{1,2,3,4\}$ and $B = \{3,4,5,6\}$. List $(A \cup B) \setminus (A \cap B)$. | $\{1, 4, 5\}$ |
| 21 | In a class of 60, 30 study French, 24 study Spanish, 20 study German. 10 study French \& Spanish, 8 study French \& German, 7 study Spanish \& German, and 4 study all three. How many study at least one language? | 53 |
| 22 | For the same class (use the totals above), how many study exactly one of the three languages? | 29 |
| 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. | $\dfrac{4}{15}$ |
| 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. | Elements that are **not** in $A$, **or** not in $B$ — equivalently, the complement of $A \cap B$. |
| 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 $y$. | $x - y = 10$ |
| 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 $n(U)$. | 49 |
| 27 | For $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$, $A = \{1, 2, 5, 6\}$, $B = \{2, 3, 6, 7\}$, find $A' \cap B$. | $\{2, 4\}$ |
| 28 | A coin and a 6-sided die are tossed together. Write the sample space $S$ as a set, and give $n(S)$. | $n(S) = 16$; e.g. $(2, 3)$ |
| 29 | Use De Morgan's laws to rewrite $(A \cup B)'$ without a union. | $A' \cup B'$ |
| 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 $x$. | $x = 5$ |
| 31 | Of 80 students, 40 like Maths, 35 like Science, 28 like Art; 15 like Maths \& Science, 10 like Maths \& Art, 8 like Science \& Art, and 4 like all three. How many like exactly one of the three subjects? | 49 |
| 32 | Verify with $U = \{1,2,3,4,5,6\}$, $A = \{1, 2, 3\}$, $B = \{3, 4, 5\}$ that $(A \cup B)' = A' \cap B'$. | Both equal $\{3, 4, 5, 6\}$. |
| 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. | $x = 6$; exactly one $= 28$ |
| 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 $n(E)$. | (a) 120; (b) 6 |
| 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? | 14 students |
| 36 | Express in set-builder notation, then in interval notation, the set of all real $x$ satisfying both $x \geq 2$ and $x < 7$. | $\{x \in \mathbb{R} : -3 < x \leq 5\} = (-3, 5]$ |
| 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? | (a) 100; (b) 70 |
| 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 $x$. | $x = 2$ |
| 39 | List all subsets of $A = \{a, b, c\}$. How many are there? | $2^4 = 16$ subsets |
| 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 $\cup$. | $\dfrac{n(T \cap G)}{n(T)} = 0.7$ |
Problèmes — Solutions détaillées
**Universal set operations.** The universal set is $U = \{1, 2, 3, \ldots, 15\}$. Let $A = \{x \in U : x \text{ is a multiple of } 3\}$ and $B = \{x \in U : x \text{ is a factor of } 12\}$. (a) List the elements of $A$, $B$, $A \cap B$, $A \cup B$, and $A'$. (b) State whether $A$ and $B$ are mutually exclusive. Justify. (c) Find $n(A' \cap B)$ and $n(A \cup B)'$.
(a) $A = \{3,6,9,12,15\}$; $B = \{1,2,3,4,6,12\}$; $A \cap B = \{3,6,12\}$; $A \cup B = \{1,2,3,4,6,9,12,15\}$; $A' = \{1,2,4,5,7,8,10,11,13,14\}$. (b) Not mutually exclusive — $A \cap B \neq \emptyset$. (c) $n(A' \cap B) = 3$; $n(A \cup B)' = 7$.
**Venn diagram from sentence.** In a class of 30 students, 18 play tennis, 12 play hockey, and 5 play both. (a) Draw a Venn diagram with all four regions labelled. (b) How many play neither tennis nor hockey? (c) Find $P(\text{plays tennis only})$. (d) Find $n(T \cup H)$ and $n((T \cup H)')$.
(a) Tennis only: 13; Both: 5; Hockey only: 7; Neither: 5. (b) 5. (c) $\frac{13}{30}$. (d) $n(T \cup H) = 25$; $n((T \cup H)') = 5$.
**Three-set Venn — school subjects.** At a school of 100 students: 55 enjoy Maths (M), 48 enjoy Science (S), 30 enjoy Art (A). 22 enjoy M & S, 10 enjoy S & A, 15 enjoy M & A, and 6 enjoy all three. (a) Use the inclusion–exclusion principle to find $n(M \cup S \cup A)$. (b) Find the number who enjoy exactly one of the three subjects. (c) Find the probability that a randomly chosen student enjoys none of the three.
(a) 92. (b) 57. (c) 0.08.
**Languages survey.** In a class of 25 students, 15 study French, 12 study Spanish, and 4 study neither. (a) How many study at least one language? (b) Use the inclusion–exclusion formula to find the number who study both. (c) Draw a Venn diagram and fill in all four regions.
(a) 21. (b) 6. (c) French only 9, both 6, Spanish only 6, neither 4.
**Sample space — two dice.** Two fair six-sided dice are rolled. (a) State the size of the sample space. (b) List the outcomes in the event $E$ = "the sum is 7". (c) List the outcomes in the event $F$ = "at least one die shows a 6". (d) Find $n(E \cap F)$ and $n(E \cup F)$.
(a) 36. (b) $\{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\}$. (c) 11 outcomes. (d) $n(E \cap F) = 2$; $n(E \cup F) = 15$.
**Three-set Venn — fitness app.** A fitness app tracks three habits among 200 users: running (R), cycling (C), swimming (S). - 80 run, 70 cycle, 60 swim. - 30 run and cycle, 25 run and swim, 20 cycle and swim. - 10 do all three. (a) Find the number who do at least one of the three activities. (b) Find the number who do none. (c) Find the number who do exactly two.
(a) 145. (b) 55. (c) 35.
**Algebraic Venn fill.** In a 2-set Venn diagram, "$A$ only" contains $2x$ elements, "$B$ only" contains $x + 5$, "both" contains $x$, and "neither" contains 4 elements. The universal set has 25 elements. (a) Set up an equation in $x$. (b) Solve for $x$. (c) State $n(A)$, $n(B)$, $n(A \cap B)$, and $n(A \cup B)$.
(a) $2x + (x + 5) + x + 4 = 25$. (b) $x = 4$. (c) $n(A) = 12$, $n(B) = 13$, $n(A \cap B) = 4$, $n(A \cup B) = 21$.
**De Morgan in action [EXT].** Let $U = \{1, 2, \ldots, 10\}$, $A = \{2, 3, 5, 7\}$, $B = \{2, 4, 6, 8, 10\}$. (a) Find $A \cup B$ and $(A \cup B)'$. (b) Find $A'$ and $B'$ and hence find $A' \cap B'$. (c) Verify your answer to (a) and (b) match De Morgan's law $(A \cup B)' = A' \cap B'$.
(a) $A \cup B = \{2,3,4,5,6,7,8,10\}$; $(A \cup B)' = \{1, 9\}$. (b) $A' = \{1,4,6,8,9,10\}$; $B' = \{1,3,5,7,9\}$; $A' \cap B' = \{1, 9\}$. (c) Both equal $\{1, 9\}$ ✓.
**Lifting from a conditional context.** In a survey, $n(U) = 50$, $n(A) = 24$, $n(B) = 20$, $n(A \cap B) = 12$. (a) Find $n(A \cup B)$ and $n(A \cup B)'$. (b) Find $n(A \cap B')$ and $n(A' \cap B)$. (c) Construct a 2-set Venn diagram showing all four regions.
(a) $n(A \cup B) = 32$; $n(A \cup B)' = 18$. (b) $n(A \cap B') = 12$; $n(A' \cap B) = 8$. (c) $A$ only 12, both 12, $B$ only 8, neither 18.
**Set-builder & interval.** Express each set in interval notation; then describe in words. (a) $\{x \in \mathbb{R} : -2 \leq x < 5\}$ (b) $\{x \in \mathbb{R} : x > 3\}$ (c) $\{x \in \mathbb{R} : 0 \leq x \leq 10 \text{ and } x \neq 5\}$
(a) $[-2, 5)$ — real numbers from $-2$ up to but not including $5$. (b) $(3, \infty)$ — reals strictly greater than $3$. (c) $[0, 5) \cup (5, 10]$ — closed interval $[0, 10]$ with the single point $5$ removed.
**Subsets and power set [EXT].** Let $A = \{a, b, c, d\}$. (a) How many subsets does $A$ have? (b) List the subsets of size 2. (c) Explain in words why a set with $n$ elements has $2^n$ subsets.
(a) 16. (b) $\{a,b\}, \{a,c\}, \{a,d\}, \{b,c\}, \{b,d\}, \{c,d\}$. (c) Each element is either "in" or "out" — two independent binary choices per element, so $2^n$ total.
**Modelling — sports club membership.** A sports club has 100 members. Each plays at least one of football (F), tennis (T), or swimming (S). 60 play F, 50 play T, 40 play S. 20 play F & T, 15 play F & S, 10 play T & S. $x$ members play all three. (a) Show that $x = 5$. (b) How many members play **exactly** one sport? (c) The treasurer wants to send a discount voucher to members who play **more than one** sport. How many vouchers are needed?
(a) See working. (b) 80. (c) 30 vouchers.