Exercises: Set notation, operations and Venn diagrams

Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, $A = \{2, 4, 6, 8, 10\}$ and $B = \{1, 2, 3, 4, 5\}$. List the elements of:

1. a) $A \cup B$
2. b) $A \cap B$
3. c) $A \setminus B$
4. d) $A'$

Using the sets from the universal set $U = \{a, b, c, d, e, f, g, h\}$, with $P = \{a, b, c, d\}$ and $Q = \{c, d, e\}$:

1. a) Find $\#P$ and $\#Q$.
2. b) Is $\{c, d\} \subset P$? Is $e \in P$? Write your answers using correct notation.
3. c) Draw a Venn diagram showing $P$ and $Q$ with all the elements placed.

In a class of 28 students, 15 play Gaelic football, 10 play camogie and 4 play both.

1. a) Draw a Venn diagram showing this information.
2. b) How many students play neither sport?

In a survey of 60 people in Limerick, 35 said they like tea, 28 like coffee and 9 like neither.

1. a) Find how many like both tea and coffee.
2. b) Draw a Venn diagram.
3. c) Find the probability that a person chosen at random likes tea only.

$U = \{x \in \mathbb{N} : 1 \le x \le 20\}$, $A$ is the set of multiples of 3, $B$ is the set of even numbers and $C$ is the set of prime numbers.

1. a) List the elements of $A$, $B$ and $C$.
2. b) Draw a three-set Venn diagram and place every element of $U$.
3. c) List $(A \cup B) \cap C'$.

Given $\#U = 40$, $\#A = 18$, $\#B = 22$ and $\#(A \cap B) = 7$:

1. a) Find $\#(A \cup B)$.
2. b) Find $\#(A \cup B)'$.
3. c) Find $\#(A \setminus B)$ and $\#(B \setminus A)$.

In a group of 100 students, 45 study French, 40 study German and 35 study Spanish. 15 study French and German, 12 study German and Spanish, 10 study French and Spanish, and 5 study all three.

1. a) Draw a three-set Venn diagram showing the number in each region.
2. b) How many study none of the three languages?
3. c) A student is chosen at random. Find the probability that they study exactly one language.

For any sets $A$ and $B$ in a universal set $U$:

1. a) Shade on Venn diagrams the sets $(A \cup B)'$ and $A' \cap B'$, and describe what you observe.
2. b) Verify that $(A \cap B)' = A' \cup B'$ for $U = \{1, \dots, 12\}$, $A = \{1, 2, 3, 4, 5, 6\}$ and $B = \{4, 5, 6, 7, 8, 9\}$.
3. c) Explain why verifying with one example does not prove the result in general.

$\#U = 50$, $\#A = 3x$, $\#B = 2x + 4$, $\#(A \cap B) = x$ and $\#(A \cup B)' = 10$.

1. a) Write an expression for $\#(A \cup B)$ in terms of $x$.
2. b) Form and solve an equation to find $x$.
3. c) Hence find $\#(A \setminus B)$ and $\#(B \setminus A)$.