Math Problem Statement

If n(U) = 100, n(X) = 51, n(Y) = 63, and n((X ∪ Y)') = 15, what is n(X ∩ Y)?

Solution

To solve this problem, let's summarize the given data and apply the formula for the union of two sets.

Given Information

  • n(U)=100n(U) = 100: the total number of elements in the universal set UU.
  • n(X)=51n(X) = 51: the number of elements in set XX.
  • n(Y)=63n(Y) = 63: the number of elements in set YY.
  • n((XY))=15n((X \cup Y)') = 15: the number of elements in the complement of XYX \cup Y.

We are asked to find n(XY)n(X \cap Y), the number of elements in the intersection of sets XX and YY.

Steps to Solution

  1. Calculate n(XY)n(X \cup Y): Since n((XY))=15n((X \cup Y)') = 15, this means that there are 15 elements outside XYX \cup Y. Therefore: n(XY)=n(U)n((XY))=10015=85n(X \cup Y) = n(U) - n((X \cup Y)') = 100 - 15 = 85

  2. Use the Formula for the Union of Two Sets: We know that: n(XY)=n(X)+n(Y)n(XY)n(X \cup Y) = n(X) + n(Y) - n(X \cap Y) Substituting the values we have: 85=51+63n(XY)85 = 51 + 63 - n(X \cap Y)

  3. Solve for n(XY)n(X \cap Y): 85=114n(XY)85 = 114 - n(X \cap Y) n(XY)=11485=29n(X \cap Y) = 114 - 85 = 29

Answer

The value of n(XY)n(X \cap Y) is 29. Thus, the correct option is:

  • a. 29

Would you like a detailed breakdown of the formula used or further examples?

Related Questions

  1. What is the concept of the universal set in set theory?
  2. How is the complement of a union calculated in set theory?
  3. What other methods could be used to find n(XY)n(X \cap Y)?
  4. How can Venn diagrams help visualize this type of problem?
  5. What are real-world applications of calculating intersections in sets?

Tip

Always verify your set notation and total counts carefully to avoid simple errors in problems involving unions and intersections.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Set Theory
Union and Intersection of Sets
Complement of a Set

Formulas

n(X ∪ Y) = n(X) + n(Y) - n(X ∩ Y)
n((X ∪ Y)') = n(U) - n(X ∪ Y)

Theorems

Principle of Inclusion-Exclusion

Suitable Grade Level

Grades 10-12