Math Problem Statement

Given n(A) = 1700, n(A ∪ B) = 2250, and n(A ∩ B) = 320, find n(B).

Solution

To find n(B)n(B), we can use the formula for the union of two sets AA and BB:

n(AB)=n(A)+n(B)n(AB)n(A \cup B) = n(A) + n(B) - n(A \cap B)

Given:

  • n(A)=1700n(A) = 1700
  • n(AB)=2250n(A \cup B) = 2250
  • n(AB)=320n(A \cap B) = 320

We need to find n(B)n(B). Plugging in the values:

2250=1700+n(B)3202250 = 1700 + n(B) - 320

Simplify the equation:

2250=1380+n(B)2250 = 1380 + n(B)

Now, solve for n(B)n(B):

n(B)=22501380=870n(B) = 2250 - 1380 = 870

So, n(B)=870n(B) = 870.

Would you like more details or have any questions?

Here are 5 related questions:

  1. What is the formula for the union of three sets?
  2. How would you calculate n(ABC)n(A \cup B \cup C) if given n(A)n(A), n(B)n(B), n(C)n(C), and the intersections?
  3. What does n(AB)=0n(A \cap B) = 0 imply about the sets AA and BB?
  4. How would you find n(AB)n(A \cup B) if n(B)n(B) was unknown but n(A)n(A) and n(AB)n(A \cap B) were given?
  5. How can Venn diagrams be used to represent these set operations?

Tip: Always double-check your work by substituting your results back into the original equation to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Union and Intersection of Sets
Basic Algebra

Formulas

n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

Theorems

Principle of Inclusion-Exclusion

Suitable Grade Level

Grades 8-10