The purpose of comparing truth tables for these two logical statements is to determine if they are equivalent, meaning that they have the same truth values for all possible combinations of truth values for the variables P, Q, and R. This comparison can help to simplify or clarify the logic of a complex statement.
The steps for constructing the truth tables for these two statements are as follows:
The parentheses in these statements indicate the order in which the logical operations should be evaluated. In P→(Q→R), the statement Q→R is evaluated first, and then the entire statement is evaluated as P→(Q→R). In (P→Q)→R, the statement P→Q is evaluated first, and then the entire statement is evaluated as (P→Q)→R. The placement of the parentheses can change the truth values and therefore the logical meaning of the statement.
If you are familiar with logical equivalences, you can use those to determine if P→(Q→R) and (P→Q)→R are equivalent. For example, you can use the associative property of implication, which states that (P→Q)→R is equivalent to P→(Q→R). If you can show that both statements are logically equivalent to a third statement, then they must also be equivalent to each other.
No, the truth tables for P→(Q→R) and (P→Q)→R will always be the same. This is because they are logically equivalent statements, meaning that they have the same truth values for all possible combinations of truth values for the variables P, Q, and R. Therefore, they will always have the same truth table regardless of the values assigned to the variables.