Math Problem Statement

Use the given information to find the number of elements in each of the regions labeled R1 through R5. n(A) = 30, n(B) = 33, n(C) = 36, n(A ∩ B) = 16, n(U) = 75

What is R5?

Solution

Let's break down the problem systematically.

We are given:

  • n(A)=30n(A) = 30 (The number of elements in set AA)
  • n(B)=33n(B) = 33 (The number of elements in set BB)
  • n(C)=36n(C) = 36 (The number of elements in set CC)
  • n(AB)=16n(A \cap B) = 16 (The number of elements in the intersection of sets AA and BB)
  • n(U)=75n(U) = 75 (The total number of elements in the universal set UU)

Steps to find R5R_5:

The regions labeled R1,R2,R3,R4,R5R_1, R_2, R_3, R_4, R_5 correspond to different parts of the Venn diagram formed by sets AA, BB, and CC. We can assume the following:

  • R1R_1 represents elements in only AA (not in BB or CC).
  • R2R_2 represents elements in only BB (not in AA or CC).
  • R3R_3 represents elements in only CC (not in AA or BB).
  • R4R_4 represents elements in ABCA \cap B \cap C (intersection of all three sets).
  • R5R_5 represents elements in neither AA, BB, nor CC (outside of all three sets).

Since R5R_5 represents elements outside of all three sets, it can be calculated using:

R5=n(U)(n(A)+n(B)+n(C)n(AB))R_5 = n(U) - (n(A) + n(B) + n(C) - n(A \cap B))

This formula accounts for the overlap between AA and BB, preventing double-counting the elements.

Let me compute this for you.It appears that using the formula as stated results in a negative value, which is not possible. This indicates a problem with the approach or the interpretation of the set intersections.

Let's reconsider the structure of the problem.

We know:

  • n(AB)=16n(A \cap B) = 16, the intersection of sets AA and BB, but we do not have specific information about intersections with CC, nor about other specific intersections (such as ACA \cap C or BCB \cap C).
  • We are looking for the number of elements outside sets AA, BB, and CC, which are counted by R5R_5.

However, the error suggests that there's missing or incorrect information about intersections, specifically about overlaps with CC. With more details on those intersections, we could correct the calculation. Would you happen to have more information on intersections involving CC?

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Math Problem Analysis

Mathematical Concepts

Set Theory
Venn Diagrams
Universal Sets
Intersections

Formulas

R5 = n(U) - (n(A) + n(B) + n(C) - n(A ∩ B))

Theorems

Principle of Inclusion-Exclusion

Suitable Grade Level

Grades 9-12