Sec 4 E-Math · Sets

Union and intersection questions with answers

Practice questions on union and intersection, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

$\xi = \{x : x \text{ is an integer and } 1 \le x \le 10\}$ , $A = \{1, 4, 9\}$ and $B = \{1, 2, 3, 4, 6, 8\}$ . List the elements of $A \cap B$ .

Step 1 of 1

∩: the elements in both A and B

A \cap B = \{1, 4\}

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Questions

Question 1List A ∩ B and A ∪ B
$\xi = \{x : x \text{ is an integer and } 1 \le x \le 12\}$ , $A = \{1, 3, 5, 7, 9, 11\}$ and $B = \{1, 2, 3, 4, 6, 12\}$ . List the elements of $A \cup B$ .
Show the worked solution
1. Step 1: ∪: every element in A or B, each listed once $A \cup B = \{1, 2, 3, 4, 5, 6, 7, 9, 11, 12\}$
Question 2List A ∩ B and A ∪ B
$\xi = \{x : x \text{ is an integer and } 1 \le x \le 18\}$ , $A = \{2, 3, 5, 7, 11, 13, 17\}$ and $B = \{1, 2, 3, 6, 9, 18\}$ . List the elements of $A \cap B$ .
Show the worked solution
1. Step 1: ∩: the elements in both A and B $A \cap B = \{2, 3\}$
Question 3The formula for n(A ∪ B)
$n(A) = 15$ , $n(B) = 6$ and $n(A \cup B) = 18$ . Find $n(A \cap B)$ .
Show the worked solution
1. Step 1: Use n(A ∪ B) = n(A) + n(B) − n(A ∩ B) $\mathstrut 18 = 15 + 6 - n(A \cap B)$ $n(A \cap B) = 3$
Question 4The formula for n(A ∪ B)
$n(A) = 21$ , $n(B) = 14$ and $n(A \cup B) = 28$ . Find $n(A \cap B)$ .
Show the worked solution
1. Step 1: Use n(A ∪ B) = n(A) + n(B) − n(A ∩ B) $\mathstrut 28 = 21 + 14 - n(A \cap B)$ $n(A \cap B) = 7$
Question 5Count from a Venn diagram
The Venn diagram shows the elements of $\xi$ , $A$ and $B$ . Find $n(A \cap B')$ .
Show the worked solution
1. Step 1: List every element of the set from the diagram $A \cap B'$ : in A but not in B. $A \cap B' = \{5, 7, 9\}$
2. Step 2: Count them $n(A \cap B') = 3$
Question 6Count from a Venn diagram
The Venn diagram shows the elements of $\xi$ , $A$ and $B$ . Find $n((A \cap B)')$ .
Show the worked solution
1. Step 1: List every element of the set from the diagram $(A \cap B)'$ : not in both. $(A \cap B)' = \{2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16\}$
2. Step 2: Count them $n((A \cap B)') = 12$

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Worked example

\xi = \{x : x \text{ is an integer and } 1 \le x \le 10\}

A = \{1, 4, 9\}

and

B = \{1, 2, 3, 4, 6, 8\}

. List the elements of

A \cap B

Step 1 of 1

∩: the elements in both A and B

A \cap B = \{1, 4\}

Questions

Question 1List A ∩ B and A ∪ B

$\xi = \{x : x \text{ is an integer and } 1 \le x \le 12\}$ , $A = \{1, 3, 5, 7, 9, 11\}$ and $B = \{1, 2, 3, 4, 6, 12\}$ . List the elements of $A \cup B$ .

Show the worked solution

Step 1: ∪: every element in A or B, each listed once $A \cup B = \{1, 2, 3, 4, 5, 6, 7, 9, 11, 12\}$

Question 2List A ∩ B and A ∪ B

$\xi = \{x : x \text{ is an integer and } 1 \le x \le 18\}$ , $A = \{2, 3, 5, 7, 11, 13, 17\}$ and $B = \{1, 2, 3, 6, 9, 18\}$ . List the elements of $A \cap B$ .

Show the worked solution

Step 1: ∩: the elements in both A and B $A \cap B = \{2, 3\}$

Question 3The formula for n(A ∪ B)

$n(A) = 15$ , $n(B) = 6$ and $n(A \cup B) = 18$ . Find $n(A \cap B)$ .

Show the worked solution

Step 1: Use n(A ∪ B) = n(A) + n(B) − n(A ∩ B) $\mathstrut 18 = 15 + 6 - n(A \cap B)$ $n(A \cap B) = 3$

Question 4The formula for n(A ∪ B)

$n(A) = 21$ , $n(B) = 14$ and $n(A \cup B) = 28$ . Find $n(A \cap B)$ .

Show the worked solution

Step 1: Use n(A ∪ B) = n(A) + n(B) − n(A ∩ B) $\mathstrut 28 = 21 + 14 - n(A \cap B)$ $n(A \cap B) = 7$

Question 5Count from a Venn diagram

The Venn diagram shows the elements of $\xi$ , $A$ and $B$ . Find $n(A \cap B')$ .

Show the worked solution

Step 1: List every element of the set from the diagram $A \cap B'$ : in A but not in B. $A \cap B' = \{5, 7, 9\}$
Step 2: Count them $n(A \cap B') = 3$

Question 6Count from a Venn diagram

The Venn diagram shows the elements of $\xi$ , $A$ and $B$ . Find $n((A \cap B)')$ .

Show the worked solution

Step 1: List every element of the set from the diagram $(A \cap B)'$ : not in both. $(A \cap B)' = \{2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16\}$
Step 2: Count them $n((A \cap B)') = 12$