Exercises: Sets and Set Operations

Let $A = \{2, 4, 6, 8, 10\}$ and $B = \{4, 8, 12, 16\}$.

1. a) List the elements of $A \cap B$.
2. b) List the elements of $A \cup B$.
3. c) List the elements of $A \setminus B$.

The universal set is $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and $A = \{1, 3, 5, 7, 9\}$.

1. a) Write down the complement $A'$ in list form.
2. b) Write $A$ in the form "rule" using set-builder notation, for example $\{x \in U \mid \dots\}$.
3. c) Write down $A \cap A'$ using the correct symbol.

Let $P = \{a, b, c, d\}$ and $Q = \{c, d, e\}$.

1. a) Is $c \in P$? Is $e \in P$? Use the symbols $\in$ or $\notin$.
2. b) Find $\#(P \cup Q)$, the number of elements in $P \cup Q$.
3. c) Is $\{c, d\} \subset P$? Explain briefly.

In a class of 30 students in Galway, 18 play camogie or hurling (set $H$), 12 play soccer (set $S$) and 5 play both.

1. a) Draw a Venn diagram description: state the number of students in $H \cap S$, in $H$ only and in $S$ only.
2. b) How many students play neither sport?
3. c) Find $\#(H \cup S)$.

$U = \{x \in \mathbb{N} \mid 1 \le x \le 12\}$, $A = \{\text{multiples of } 3\}$ and $B = \{\text{even numbers}\}$.

1. a) List $A$, $B$, $A \cap B$ and $A \cup B$.
2. b) List $(A \cup B)'$.
3. c) List $B \setminus A$.

$U = \{1, 2, 3, \dots, 10\}$, $A = \{1, 2, 3, 4, 5\}$, $B = \{4, 5, 6, 7\}$ and $C = \{5, 7, 9, 10\}$.

1. a) List $A \cap B \cap C$.
2. b) List $(A \cup B) \cap C$.
3. c) List $A \setminus (B \cup C)$.
4. d) Verify that $A \cup (B \cup C) = (A \cup B) \cup C$.

A survey of 60 tourists in Kerry found that 35 visited the Ring of Kerry ($R$), 28 visited Killarney National Park ($K$) and 22 visited Dingle ($D$). Also, 12 visited $R$ and $K$ only, 9 visited $R$ and $D$ only, 7 visited $K$ and $D$ only, and 4 visited all three.

1. a) Find the number who visited $R$ only, $K$ only and $D$ only.
2. b) Find how many visited none of the three places.
3. c) Is the data consistent? Justify your answer.

For sets $A$ and $B$ in a universal set $U$, $\#(A) = 20$, $\#(B) = 15$ and $\#(A \cup B) = 28$.

1. a) Find $\#(A \cap B)$.
2. b) Find $\#(A \setminus B)$ and $\#(B \setminus A)$.
3. c) If $\#(U) = 40$, find $\#((A \cup B)')$.
4. d) Explain why $\#(A \cap B)$ could not be 16.

Let $U = \{1, 2, \dots, 20\}$, $A = \{\text{prime numbers}\}$, $B = \{\text{odd numbers}\}$ and $C = \{\text{multiples of } 5\}$.

1. a) List $A$, $B$ and $C$.
2. b) Find $A \cap B \cap C$ and $(A \cap B) \setminus C$.
3. c) Show, using these sets, that $(A \cup B)' = A' \cap B'$.
4. d) Which element of $U$ is in $A$ but not in $B$? Explain why it is the only one.