Question

For two sets \( A \) and \( B \), we have \( n(A \cup B) = 50 \), \( n(A \cap B) = 12 \), and \( n(A - B) = 15 \). Then \( n(B - A) \) is equal to:

• A. 27
• B. 35
• C. 38
• D. 29
• E. 23

Hint:

When dealing with set theory problems, use the formulas for union and intersection to relate the various set operations. Be careful with how you express the terms for each set and always check the given values.

Answer 1

Answer: (E)

We are given the following information: \[ n(A \cup B) = 50, \quad n(A \cap B) = 12, \quad n(A - B) = 15. \]
We need to find \( n(B - A) \).
Using the principle of set theory, the number of elements in the union of two sets can be expressed as: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B). \] Also, we know that: \[ n(A) = n(A - B) + n(A \cap B), \quad n(B) = n(B - A) + n(A \cap B). \] Substituting the known values into the equations: \[ 50 = n(A) + n(B) - 12, \] where \[ n(A) = 15 + 12 = 27 \quad {(since } n(A - B) = 15 { and } n(A \cap B) = 12 {)}. \] Now substitute \( n(A) = 27 \) into the first equation: \[ 50 = 27 + n(B) - 12 \quad \Rightarrow \quad n(B) = 35. \] Next, we calculate \( n(B - A) \): \[ n(B) = n(B - A) + n(A \cap B) \quad \Rightarrow \quad 35 = n(B - A) + 12 \quad \Rightarrow \quad n(B - A) = 23. \] Thus, the number of elements in \( B - A \) is \( \boxed{23} \).