Exercises: Venn diagrams with two sets

In a class of $20$ students in Galway, $12$ play camogie or hurling (set $A$), and $8$ of those $12$ also play football. Describe how a two-set Venn diagram would show $A \cap B$ if $B$ is the set of students who play football, and state $\#(A \cap B)$.

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

1. a) Place each element of $U$ in the correct region of a Venn diagram: only $A$, both, only $B$, neither.

1. b) State $\#A'$.

In a Venn diagram, $\#(A \cap B) = 5$, $\#(A \text{ only}) = 7$, $\#(B \text{ only}) = 4$ and $\#(\text{neither}) = 6$.

1. a) Find $\#A$.

1. b) Find $\#B$.

1. c) Find $\#U$.

In a 1st Year class of $28$ students in Cork, $15$ have a dog, $11$ have a cat and $6$ have both.

1. a) Draw or describe the Venn diagram, giving the number in each region.

1. b) How many have neither a dog nor a cat?

1. c) How many have a dog but no cat?

Of $40$ students in a school in Limerick, $24$ study French (F) and $18$ study German (G). $5$ study neither.

1. a) Find $\#(F \cup G)$.

1. b) Find $\#(F \cap G)$.

1. c) How many study French only?

$U = \{x \mid x \in \mathbb{N}, \ x \leq 12\}$, $A$ is the set of even numbers and $B$ is the set of multiples of $3$.

1. a) Draw or describe the Venn diagram, with all elements placed.

1. b) List $(A \cup B)'$.

1. c) List $A \setminus B$.

In a survey of $60$ people at a Dublin train station, $x$ travelled by Luas, $2x$ by DART, $10$ by both and $7$ by neither. Every person in the survey is in $U$.

1. a) Write an equation in $x$ using the number of people in each region.

1. b) Solve it.

1. c) How many used the DART only?

In a club of $50$ members, $30$ like swimming (S) and $25$ like cycling (C). Let $n$ be the number who like both.

1. a) Find the smallest and the largest possible values of $n$, justifying each.

1. b) If $4$ members like neither, find $n$.

$A$ and $B$ are subsets of $U$ with $\#U = 36$, $\#A = 20$, $\#B' = 18$ and $\#(A' \cap B') = 6$.

1. a) Find $\#B$.

1. b) Find $\#(A \cup B)$.

1. c) Find $\#(A \cap B)$.

1. d) Find $\#(A \setminus B)$.