Question: "(P^~Q)^(P→Q) check whether it is in contradiction..."

College
Algebra
Linear algebra
Vectors and spaces

Answered question

2021-11-12

(P^~Q)^(P→Q)

check whether it is in contradiction form or not.

Answer & Explanation

user_27qwe

Skilled2023-04-21Added 375 answers

To check whether this expression is in contradiction form or not, we need to simplify it using the laws of logic. First, let's use the definition of the conditional to rewrite $P \to Q$ as $~ P \lor Q$ (not P or Q):
${(P^{~ Q})}^{~ P \lor Q}$

Next, we can use De Morgan's laws to distribute the negation over the parentheses:
${(P^{~ Q})}^{~} P \land {(P^{~ Q})}^{Q}$

Now we can simplify the first part (P^(~Q))^~P using the law of contradiction, which states that $P^{~ P}$ is always false (contradictory):
${(P^{~ Q})}^{Q}$

We can simplify this expression further using the commutative property of conjunction:
$Q^{P^{~ Q}}$

This expression is not in contradiction form, because it is not in the form $P^{~ P}$ . Therefore, we have shown that ${(P^{~ Q})}^{P \to Q}$ is not a contradiction.

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