11.1 Sets and Venn Diagrams

**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 $n(A \cup B)'$.

**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 $n((T \cup H)')$.

**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.

**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.

**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 $n(E \cup F)$.

**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.

**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 $n(A \cup B)$.

**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 $(A \cup B)' = A' \cap B'$.

**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.

**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) $\{x \in \mathbb{R} : 0 \leq x \leq 10 \text{ and } x \neq 5\}$

**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.

**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?