FORM GR1268 Question 16

Question:
Suppose \(A, B,\) and \(C\) are statements such that \(C\) is true if exactly one of \(A\) and \(B\) is true. If \(C\) is false, which of the following statements must be true?

(A) If \(A\) is true, then \(B\) is false.
(B) If \(A\) is false, then \(B\) is false.
(C) If \(A\) is false, then \(B\) is true.
(D) Both \(A\) and \(B\) are true.
(E) Both \(A\) and \(B\) are false.

Answer:
(B)

Answer Key:
There are four scenarios:
\(A\) is true and \(B\) is true \(\Rightarrow\) \(C\) is false.
\(A\) is true and \(B\) is false \(\Rightarrow\) \(C\) is true.
\(A\) is false and \(B\) is true \(\Rightarrow\) \(C\) is true.
\(A\) is false and \(B\) is false \(\Rightarrow\) \(C\) is false.

If \(C\) is false, then either (i) both \(A\) and \(B\) are true; or (ii) both \(A\) and \(B\) are false.

Thus, if one is true, the other must be true.
If one is false, the other must be false.

(D) is not necessarily true because both (A) and (B) can be false.
(E) is not necessarily true because both (A) and (B) can be true.