----------------------------------------- Write the negation of each of the foll

Question

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Write the negation of each of the following logical expressions so that all negations immediately precede predicates. In some cases, it may be necessary to apply one or more laws of propositional logic.

(d)

x y (P(x, y) P(y, x))

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Which of the following arguments are invalid and which are valid? Prove your answer by replacing each proposition with a variable to obtain the form of the argument. Then prove that the form is valid or invalid.

If 22 is an irrational number, then 2222 is an irrational number.

2222 is an irrational number.

22 is an irrational number.

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Prove that each argument is valid by replacing each proposition with a variable to obtain the form of the argument. Then use the rules of inference to prove that the form is valid.

(a)

If I drive on the freeway, I will see the fire.

I will drive on the freeway or take surface streets (or both).

I am not going to take surface streets.

I will see the fire.

Explanation / Answer

d)
¬(x y (P(x, y) P(y, x)))
= x ¬(y (P(x, y) P(y, x)))
= x y ¬(P(x, y) P(y, x))
= x y ¬(P(x, y) P(y, x) V P'(x, y) P'(y, x) )
=x y P'(x, y) P(y,x) V P(x, y) P'(y,x)

I hope this answer helps you. please let me know if you need more help :)

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