Use the union rule to answer the question.
If [tex]$n(A)=9, n(B)=12$[/tex], and [tex]$n(A \cap B)=3$[/tex], what is [tex]$n(A \cup B)$[/tex]?
[tex]$n(A \cup B)= \square$[/tex] (Type a whole number.)
Answer (1)
The problem gives the number of elements in set A, set B, and their intersection.
The union rule states: n ( A ∪ B ) = n ( A ) + n ( B ) − n ( A ∩ B ) .
Substituting the given values: n ( A ∪ B ) = 9 + 12 − 3 .
Calculating the result gives the number of elements in A ∪ B : 18 .
Explanation
Understand the problem and provided data We are given the number of elements in set A, n ( A ) = 9 . We are given the number of elements in set B, n ( B ) = 12 . We are given the number of elements in the intersection of sets A and B, n ( A ∩ B ) = 3 . We are asked to use the union rule to find the number of elements in the union of sets A and B, n ( A ∪ B ) .
State the union rule The union rule states that for any two sets A and B, the number of elements in the union of A and B is given by:
n ( A ∪ B ) = n ( A ) + n ( B ) − n ( A ∩ B )
Substitute the given values Now, we substitute the given values into the formula:
n ( A ∪ B ) = 9 + 12 − 3
Calculate the result Performing the addition and subtraction, we get:
n ( A ∪ B ) = 21 − 3 = 18
State the final answer Therefore, the number of elements in the union of sets A and B is 18.
Examples
Understanding set unions is crucial in many real-world scenarios. For example, if you're planning a party and need to know how many unique guests are coming from two different invitation lists, you use the union rule. Suppose list A has 9 people, list B has 12 people, and 3 people are on both lists. The total number of unique guests is calculated as 9 + 12 − 3 = 18 . This ensures you don't over-prepare for guests who were invited twice!
