Three agricultural engineers ($X, Y, Z$) inspect a farm drainage system. It is known that:
- If $X$ finds a defect, then $Y$ also finds a defect.
- Exactly one of $Y$ or $Z$ finds a defect, but not both.
- $Z$ did not find any defect.
Which of the following statements is/are logically TRUE?
Options:
(A) $Y$ found a defect.
(B) $X$ definitely found a defect.
(C) It is possible that $X$ did not find any defect.
(D) Both $X$ and $Z$ found defects.
Answer Key:
A, C
Step-by-Step Solution & Explanation
**Method 1: Step-by-Step Truth Deduction**
1. Fact: $Z$ did not find a defect ($Z = \text{False}$).
2. Premise: 'Exactly one of $Y$ or $Z$ finds a defect' ($Y \oplus Z = \text{True}$).
Since $Z = \text{False}$, we must have $Y = \text{True}$ ($Y$ found a defect). Thus (A) is TRUE.
3. Premise: 'If $X$ finds a defect, then $Y$ finds a defect' ($X \implies Y$).
Since $Y = \text{True}$, the conditional $X \implies Y$ is satisfied whether $X$ is True or False.
Therefore, $X$ may or may not have found a defect; it is not guaranteed that $X$ found one. Thus (B) is false and (C) is TRUE.
4. Since $Z$ did not find a defect, (D) is false.
Therefore, options **A** and **C** are strictly correct.