If n(A U B) = 99 and n(A) n(B) = 61, find n(A N B).

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
The given mathematical expression is:

"If \( n(A \cup B) = 99 \) and \( n(A) = n(B) = 61 \), find \( n(A \cap B) \)."

### Explanation:

This expression involves set theory. Here, \( n(A \cup B) \) represents the number of elements in the union of sets A and B, which is given as 99. The number of elements in set A, \( n(A) \), and in set B, \( n(B) \), are both given as 61. The goal is to determine \( n(A \cap B) \), which is the number of elements common to both sets A and B.

### Solution:

To find \( n(A \cap B) \), we can use the formula for the union of two sets:

\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\]

Substitute the given values into the equation:

\[
99 = 61 + 61 - n(A \cap B)
\]

Simplify:

\[
99 = 122 - n(A \cap B)
\]

Solve for \( n(A \cap B) \):

\[
n(A \cap B) = 122 - 99
\]

\[
n(A \cap B) = 23
\]

Thus, the number of elements common to both sets A and B is 23.
Transcribed Image Text:The given mathematical expression is: "If \( n(A \cup B) = 99 \) and \( n(A) = n(B) = 61 \), find \( n(A \cap B) \)." ### Explanation: This expression involves set theory. Here, \( n(A \cup B) \) represents the number of elements in the union of sets A and B, which is given as 99. The number of elements in set A, \( n(A) \), and in set B, \( n(B) \), are both given as 61. The goal is to determine \( n(A \cap B) \), which is the number of elements common to both sets A and B. ### Solution: To find \( n(A \cap B) \), we can use the formula for the union of two sets: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] Substitute the given values into the equation: \[ 99 = 61 + 61 - n(A \cap B) \] Simplify: \[ 99 = 122 - n(A \cap B) \] Solve for \( n(A \cap B) \): \[ n(A \cap B) = 122 - 99 \] \[ n(A \cap B) = 23 \] Thus, the number of elements common to both sets A and B is 23.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,