Determine whether the following compound statement is a tautology or contradiction.(p → (q → r)) → ((p ∧ q ) → r)
Question
Determine whether the following compound statement is a tautology or contradiction.(p → (q → r)) → ((p ∧ q ) → r)
Solution
To determine whether the compound statement is a tautology or contradiction, we need to construct a truth table. Here are the steps:
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Identify the individual statements: In this case, we have three individual statements: p, q, and r.
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Construct a truth table: This table should have a row for every possible combination of truth values for the individual statements. Since we have three statements, we will have 2^3 = 8 rows.
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Determine the truth values for the compound statements: We do this by applying the logical operators in the compound statement to the truth values of the individual statements.
Here is the truth table:
| p | q | r | q → r | p → (q → r) | p ∧ q | (p ∧ q) → r | (p → (q → r)) → ((p ∧ q ) → r) |
|---|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T | T |
| T | T | F | F | F | T | F | T |
| T | F | T | T | T | F | T | T |
| T | F | F | T | T | F | T | T |
| F | T | T | T | T | F | T | T |
| F | T | F | F | T | F | T | T |
| F | F | T | T | T | F | T | T |
| F | F | F | T | T | F | T | T |
As you can see, the final column (which represents the entire compound statement) is true in all cases. Therefore, the compound statement is a tautology.
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