Set notation questions with answers

Questions

1. Question 1List the elements

   $\xi = \{x : x \text{ is an integer and } 1 \le x \le 12\}$ and $A$ is the set of multiples of 4 in $\xi$. List the elements of $A$.

   Show the worked solution

   1. Step 1: List the multiples of 4 from 1 to 12$A = \{4, 8, 12\}$
2. Question 2Number of elements

   $\xi = \{x : x \text{ is an integer and } 1 \le x \le 15\}$ and $A$ is the set of integers less than 6 in $\xi$. Find $n(A)$.

   Show the worked solution

   1. Step 1: List the integers less than 6 from 1 to 15$A = \{1, 2, 3, 4, 5\}$
   2. Step 2: n(A) is the number of elements$n(A) = 5$
3. Question 3True or false: ∈ and ∉

   $\xi = \{x : x \text{ is an integer and } 2 \le x \le 16\}$ and $A$ is the set of multiples of 5 in $\xi$. State whether $6 \in A$ is true or false.

   Show the worked solution

   1. Step 1: List the multiples of 5 from 2 to 16$A = \{5, 10, 15\}$
   2. Step 2: Is the number in your list for A?$6 \notin A$So the statement is false.
4. Question 4The complement

   $\xi = \{x : x \text{ is an integer and } 1 \le x \le 12\}$ and $A$ is the set of factors of 20 in $\xi$. List the elements of $A'$.

   Show the worked solution

   1. Step 1: List the factors of 20 from 1 to 12$A = \{1, 2, 4, 5, 10\}$
   2. Step 2: A′ is everything in ξ that is not in A$A' = \{3, 6, 7, 8, 9, 11, 12\}$
5. Question 5A universal set with an unknown end k

   $\xi = \{x : x \text{ is an integer and } 7 \le x \le k\}$ and $A$ is the set of multiples of 5 in $\xi$. Given that $n(A) = 3$, find the least possible value of $k$.

   Show the worked solution

   1. Step 1: The multiples of 5 from 7 upwards$10, 15, 20, \ldots$
   2. Step 2: n(A) = 3: ξ must reach the 3th multiple, so the least k is that multiple$k = 20$