To create a Truth Table for the statement ~(P v (~P ∧ Q)) → R, we need to list all possible combinations of truth values for P, Q, and R and then evaluate the truth value of the entire statement for each combination.
Now, we can use the Truth Table to prove the statement using the Converse, Inverse, and Contrapositive.
Converse: R → ~(P v (~P ∧ Q))
To prove the converse, we need to assume R and show that ~(P v (~P ∧ Q)) must also be true. From the Truth Table, we can see that when R is true, ~(P v (~P ∧ Q)) is also true in all cases. Therefore, the converse is true.
Inverse: ~(~P v (~P ∧ Q)) → ~R
To prove the inverse, we need to assume ~(~P v (~P ∧ Q)) and show that ~R must also be true. From the Truth Table, we can see that when ~(~P v (~P ∧ Q)) is true, ~R is also true in all cases. Therefore, the inverse is true.
Contrapositive: ~R → ~(~P v (~P ∧ Q))
To prove the contrapositive, we need to assume ~R and show that ~(~P v (~P ∧ Q)) must also be true. From the Truth Table, we can see that when ~R is true, ~(~P v (~P ∧ Q)) is also true in all cases. Therefore, the contrapositive is true.
Conclusion: We have shown that the original statement ~(P v (~P ∧ Q)) → R is true, as well as its converse, inverse, and contrapositive
Answers & Comments
Answer:
To create a Truth Table for the statement ~(P v (~P ∧ Q)) → R, we need to list all possible combinations of truth values for P, Q, and R and then evaluate the truth value of the entire statement for each combination.
Now, we can use the Truth Table to prove the statement using the Converse, Inverse, and Contrapositive.
Converse: R → ~(P v (~P ∧ Q))
To prove the converse, we need to assume R and show that ~(P v (~P ∧ Q)) must also be true. From the Truth Table, we can see that when R is true, ~(P v (~P ∧ Q)) is also true in all cases. Therefore, the converse is true.
Inverse: ~(~P v (~P ∧ Q)) → ~R
To prove the inverse, we need to assume ~(~P v (~P ∧ Q)) and show that ~R must also be true. From the Truth Table, we can see that when ~(~P v (~P ∧ Q)) is true, ~R is also true in all cases. Therefore, the inverse is true.
Contrapositive: ~R → ~(~P v (~P ∧ Q))
To prove the contrapositive, we need to assume ~R and show that ~(~P v (~P ∧ Q)) must also be true. From the Truth Table, we can see that when ~R is true, ~(~P v (~P ∧ Q)) is also true in all cases. Therefore, the contrapositive is true.
Conclusion: We have shown that the original statement ~(P v (~P ∧ Q)) → R is true, as well as its converse, inverse, and contrapositive
Step-by-step explanation:
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