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Chapter 6: Problem 13

If \(n(A)=4, n(B)=5\), and \(n(A \cup B)=9\), find \(n(A \cap B)\).

Short Answer

Expert verified
The number of elements in the intersection of sets A and B, denoted as \(n(A \cap B)\), is 0.

Step by step solution

01

Identify given information

We are given three pieces of information about sets A and B: 1. The number of elements in set A is 4, denoted as \(n(A) = 4\). 2. The number of elements in set B is 5, denoted as \(n(B) = 5\). 3. The number of elements in the union of A and B is 9, denoted as \(n(A \cup B) = 9\).
02

Apply the Principle of Inclusion and Exclusion

The Principle of Inclusion and Exclusion states that: \[n(A \cup B) = n(A) + n(B) - n(A \cap B)\] Where \(n(A \cap B)\) is the number of elements in the intersection of A and B. Our goal is to find the value of \(n(A \cap B)\).
03

Substitute the given information

Using the given information, substitute the values of \(n(A)\), \(n(B)\), and \(n(A \cup B)\) into the equation: \[9 = 4 + 5 - n(A \cap B)\]
04

Solve for n(A 鈭� B)

Now, solve for \(n(A \cap B)\): \[9 = 9 - n(A \cap B)\] Subtract 9 from both sides of the equation: \[0 = -n(A \cap B)\] Multiply both sides by -1: \[0 = n(A \cap B)\] So, the number of elements in the intersection of sets A and B, denoted as \(n(A \cap B)\), is 0.

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Most popular questions from this chapter

Let \(U=\\{1,2,3,4,5,6,7, a, b, c, d, e\\} .\) If \(A=\\{1,2, a, e)\) and \(B=\\{1,2,3,4, a, b, c\\}\), find: a. \(n\left(A^{c}\right)\) b. \(n\left(A \cap B^{\circ}\right)\) c. \(n\left(A \cup B^{\circ}\right)\) d. \(n\left(A^{c} \cap B^{c}\right)\)

Computers manufactured by a certain company have a serial number consisting of a letter of the alphabet followed by a four-digit number. If all the serial numbers of this type have been used, how many sets have already been manufactured?

Verify the equation $$ n(A \cup B)=n(A)+n(B) $$ for the given disjoint sets. $$ \begin{array}{l} A=|x| x \text { is a whole number between } 0 \text { and } 4\\} \\ B=|x| x \text { is a negative integer greater than }-4\\} \end{array} $$

If \(n(A)=12, n(B)=12, n(A \cap B)=5, n(A \cap C)=5\), \(n(B \cap C)=4, n(A \cap B \cap C)=2\), and \(n(A \cup B \cup C)=\) 25, find \(n(C) .\)

Find the smallest possible set (i.e.. the set with the least number of elements) that contains the given sets as subsets. $$ \\{1,2\\},\\{1,3,4\\},\\{4,6,8,10\\} $$
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