Problem 1. (25 points) Convert each of the following statements to an implicatio

Question

Problem 1. (25 points) Convert each of the following statements to an implication in propositional logic, and prove that the statement is true by proving that the implication is a tautology. (a) If you are sane, you are not dreaming if you can read this question. Therefore, if you are dreaming, you are insane if you can read this question. (b) If both Romeo and Juliet learn of each other's plan, then neither of them will die. If Romeo learns of Juliet's plan, but Juliet does not learn of Romeo's, then only Juliet will die, and vice versa. Therefore, if they both died, neither learned of the other's plan.

Explanation / Answer

--->a) suppose if P is:you are sane;

Q:is you are dreaming;

S:is you can read this question;

so the follwing is the propositional logic

((P and ~Q)--->R)------->((Q and ~P)--->R)

suppose let us take the labales x as (P and ~Q)--->R)

y as(Q and ~P)--->R

SO FROM THE ABOVE TABLE THE LAST COLUMN CONTAINS ALL THE FEILDS AS TRUE SO WE CAN SAY THAT IT IS A TAUTOLOGY

------>b)similarly here also we can label the propositions as follows:

P: Romeo learns the Juleit plan

Q: Juleit learns the Romeo's Plan

R:one of them will die;

~R: neither of them will die

we can write the implications as follows

[((P and Q)--->~R)] and [((P and ~Q)--->R) and ((~P and Q)--->R)] ---->(logically implies)(R---->(~P and ~Q))

P Q R ~Q ~P P and ~Q Q and ~P (P and ~Q )---> R (Q and ~P)--->R x----->y T T T F F T T T T T T T F F F T T F F T T F T T F F F T T T T F F T F F F T T T F T T F T F F T T T F T F F T F F T T T F F T T T F F T T T F F F T T F F T T T

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