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Read the following premises:

  1. $\textbf{A} \rightarrow X \text{ or } Y \text{ or not }P$
  2. $\textbf{B} \rightarrow \text{ not }Y$
  3. $A \text{ or not } B \rightarrow P$
  4. $Y \text{ and } A \text{ are true}$

Which of the following statements is correct?

  1. $X$ and $Y$ are both true.
  2. Only $X$ is true.
  3. $\textbf{A}$ and $\textbf{B}$ are false.
  4. Either $X$ or $Y$ is true.
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A and Y is true means A.Y is true. It can only be possible when A is true and Y is true. So
1. A
2. A --> (X v Y v ~P)

From 1 and 2 By using modus ponen we have this.
3. (X v Y v ~P)
Now,
4. Y
5. B --> ~Y
From 4 and 5 by using modus tollen we have this.
6. ~B.

So A is true and B is false. Hence A.B is false. So option C should be correct.

For option D we already know that Y is true. Hence (X v Y) is also true. So option D is also correct.
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