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Chapter 6: Problem 8

If \(n(A)=10, n(A \cup B)=15\), and \(n(B)=8\), then what is \(n(A \cap B) ?\)

Short Answer

Expert verified
\(n(A \cap B) = 3\)

Step by step solution

01

Write down the formula for union of two sets

Recall the formula for the union of two sets A and B: \( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)
02

Plug in the given values

Now, plug in the given values for \(n(A)\), \(n(A \cup B)\), and \(n(B)\) into the formula: \( 15 = 10 + 8 - n(A \cap B) \)
03

Solve for the intersection

Solve the equation for \(n(A \cap B)\): \( 15 = 18 - n(A \cap B) \) Subtract 18 from both sides of the equation: \( -3 = -n(A \cap B) \) Now, multiply both sides by -1 to find the value of \(n(A \cap B)\): \( 3 = n(A \cap B) \)
04

State the answer

The number of elements in the intersection of sets A and B is 3. Therefore, \(n(A \cap B) = 3\).

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Most popular questions from this chapter

In a survey of 200 members of a local sports club, 100 members indicated that they plan to attend the next Summer Olympic Games, 60 indicated that they plan to attend the next Winter Olympic Games, and 40 indicated that they plan to attend both games. How many members of the club plan to attend a. At least one of the two games? b. Exactly one of the games? c. The Summer Olympic Games only? d. None of the games?

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