11.1 Sets and Venn Diagrams

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 $A \cap B$.

Using $A = \{ 1,2,3 \}$ and $B = \{ 2,3,4,5 \}$, list $A \cup B$.

$U = \{1,2,3,4,5,6,7,8,9,10\}$ and $A = \{ 2,4,6,8,10 \}$. List $A'$.

In a Venn diagram, $n(A) = 12$, $n(B) = 9$ and $n(A \cap B) = 4$. Find $n(A \cup B)$.

In a class of 30 students, 18 play tennis, 12 play hockey, and 5 play both. How many play neither?

Let $A = \{ 2,4,6,8,10 \}$. State whether $ 5 \in A$ or $ 5 \notin A$.

Let $A = \{2, 4, 6\}$ and $B = \{ 2,4,6,8,10 \}$. State whether $A \subseteq B$ is true or false.

List the elements of $\{x \in \mathbb{Z} : 1 \leq x \leq 5\}$.

Let $A = \{ 1,3,5 \}$ and $B = \{ 2,4,6 \}$. Find $A \cap B$.

In a universal set $U$ with $n(U) = 50$, $n(A) = 18$. Find $n(A')$.

In a survey of 50 people, 30 like tea, 24 like coffee, and 8 like neither. How many like both?

In a class of 30, 18 study French, 12 study Spanish and 7 study both. How many study only French?

$U = \{1, 2, \ldots, 15\}$, $A = \{$multiples of 3$\}$, $B = \{$factors of 12$\}$. Find $A \cap B$.

$n(A) = 22$, $n(B) = 18$, $n(C) = 15$. 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 $n(A \cup B \cup C)$.

Write the set $\{x \in \mathbb{R} : -2 \leq x < 5\}$ in interval notation.

$U = \{1, 2, 3, 4, 5\}$, $A = \{1, 3, 5\}$, $B = \{2, 3\}$. List $(A \cup B)'$.

Are $A = \{$letters in MATHS$\}$ and $B = \{$letters in STAMH$\}$ equal sets?

Given $n(U) = 50$, $n(A) = 30$, $n(B) = 20$ and $n(A \cup B)' = 8$, find $n(A \cap B)$.

Two fair dice are rolled. State the size of the sample space and list the outcomes giving a sum of 5.

Let $A = \{1,2,3,4\}$ and $B = \{3,4,5,6\}$. List $(A \cup B) \setminus (A \cap B)$.

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?

For the same class (use the totals above), how many study exactly one of the three languages?

$U = \{1, 2, \ldots, 15\}$, $A = \{$multiples of 3$\}$, $B = \{$factors of 12$\}$. Find $P(A' \cap B)$ when an element is chosen at random.

In a 2-set Venn diagram for $A$ and $B$, describe in words the region $A \cap B'$, and give the number-elements interpretation.

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 $y$.

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 $n(U)$.

For $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$, $A = \{1, 2, 5, 6\}$, $B = \{2, 3, 6, 7\}$, find $A' \cap B$.

A coin and a 6-sided die are tossed together. Write the sample space $S$ as a set, and give $n(S)$.

Use De Morgan's laws to rewrite $(A \cup B)'$ without a union.

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 $x$.

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?

Verify with $U = \{1,2,3,4,5,6\}$, $A = \{1, 2, 3\}$, $B = \{3, 4, 5\}$ that $(A \cup B)' = A' \cap B'$.

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.

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 $n(E)$.

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?

Express in set-builder notation, then in interval notation, the set of all real $x$ satisfying both $x \geq 2$ and $x < 7$.

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?

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 $x$.

List all subsets of $A = \{a, b, c\}$. How many are there?

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 $\cup$.