Comparing Truth Tables: P→(Q→R) vs. (P→Q)→R

  • Thread starter Mdhiggenz
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In summary, the purpose of comparing truth tables for P→(Q→R) and (P→Q)→R is to determine their equivalence. The steps for constructing the truth tables involve listing all possible combinations of truth values and evaluating the statements. The parentheses in these statements indicate the order of evaluation. To determine equivalence without a truth table, you can use logical equivalences. The truth tables for these statements will always be the same due to their logical equivalence.
  • #1
Mdhiggenz
327
1

Homework Statement



Hello, I was working on the following truth table problems
1.P→(Q→R)
2.(P→Q)→R

and wanted to know why I got different truth tables it seems that we could use the associative law to rewrite 1. in the same manner as 2.

Or am I missing something?

Thanks

Higgenz




Homework Equations





The Attempt at a Solution

 
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  • #2
What "associative law" are you talking about? As far as I know there is no "associative law" for implications.

Here, these two statements are clearly not the same.
 

FAQ: Comparing Truth Tables: P→(Q→R) vs. (P→Q)→R

What is the purpose of comparing truth tables for P→(Q→R) and (P→Q)→R?

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.

What are the steps for constructing the truth tables for P→(Q→R) and (P→Q)→R?

The steps for constructing the truth tables for these two statements are as follows:

  1. List all possible combinations of truth values for the variables P, Q, and R.
  2. For each combination, evaluate the innermost statement in parentheses first, and then evaluate the outer statement.
  3. Record the resulting truth values in the final column of the truth table.
  4. Compare the truth values in the final column for both statements to determine if they are equivalent.

What is the significance of the parentheses in P→(Q→R) and (P→Q)→R?

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.

How can I determine if P→(Q→R) and (P→Q)→R are equivalent without constructing a truth table?

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.

Can the truth tables for P→(Q→R) and (P→Q)→R ever be different?

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.

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