Exercises: Set notation and listing elements

Let $A$ be the set of the months of the year that begin with the letter J.

1. a) List the elements of $A$.

1. b) Write down $\#A$.

1. c) Is March $\in A$? Answer "yes" or "no".

Let $B = \{x \mid x \text{ is an even number}, \ 3 < x \leq 15\}$.

1. a) List the elements of $B$.

1. b) Write down $\#B$.

1. c) Fill in the symbol $\in$ or $\notin$: $8 \ \square \ B$ and $9 \ \square \ B$.

Let $C = \{2, 4, 6, 8, 10\}$ and $D = \{4, 8\}$.

1. a) Is $D \subset C$? Answer "yes" or "no".

1. b) Is $C \subset D$? Answer "yes" or "no".

1. c) Let $E$ be the set of whole numbers greater than $10$ and less than $11$. Write $E$ using the correct symbol.

The universal set is $U = \{1, 2, 3, \ldots, 20\}$. Let $P$ be the set of prime numbers in $U$ and let $S$ be the set of square numbers in $U$.

1. a) List the elements of $P$ and of $S$.

1. b) Write down $P \cap S$ and say what this tells you.

1. c) Write down $\#(P \cup S)$.

Let $A = \{1, 3, 5, 7, 9, 11\}$ and $B = \{3, 6, 9, 12\}$, with universal set $U = \{1, 2, 3, \ldots, 12\}$.

1. a) List $A \cap B$ and $A \cup B$.

1. b) List $A \setminus B$ and $B \setminus A$.

1. c) List $A'$, the complement of $A$.

Write down all the subsets of $\{a, b, c\}$, including the empty set $\emptyset$.

1. a) List them.

1. b) How many subsets are there?

1. c) How many subsets would a set with $4$ elements have?

The universal set is $U = \{x \mid x \in \mathbb{N}, \ 1 \leq x \leq 30\}$. Let $A$ be the multiples of $3$ and $B$ the multiples of $4$ in $U$.

1. a) List $A \cap B$.

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

1. c) Show that $\#(A \cup B) = \#A + \#B - \#(A \cap B)$ for these sets.

Sets $A$ and $B$ satisfy $\#A = 8$, $\#B = 11$ and $\#(A \cap B) = 3$. The universal set $U$ has $25$ elements.

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

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

1. c) Find $\#(A \setminus B)$ and $\#(B \setminus A)$.

1. d) Explain why $\#(A \cap B)$ can never be greater than $8$ for these sets.

Let $A = \{1, 2, 3, 4, 5, 6\}$. The set $B$ is a subset of $A$ with $\#B = 4$. The set $C$ satisfies $B \cap C = \{2, 5\}$ and $B \cup C = \{1, 2, 3, 4, 5, 6, 7\}$, and $3 \notin C$.

1. a) Find $C$ if $1 \in B$ and $4 \notin B$.

1. b) Hence list $B$.

1. c) Justify why $7 \in C \setminus B$.