Propositional Logic: Tautologies and Contradictions

A tautology is a propositional formula that is always true, regardless of the truth values of its constituent propositions. In this case, the propositional formula "p → (p ∨ q)" is always true, because if "p" is true, then "p ∨ q" is also true, and if "p" is false, then "p → (p ∨ q)" is also true.

A contradiction is a propositional formula that is always false, regardless of the truth values of its constituent propositions. In this case, the propositional formula "(p ∨ q)" is always false, because if "p" is true or "q" is true, then "p ∨ q" is true, and "(p ∨ q)" is false.

A tautology is a propositional formula that is always true, regardless of the truth values of its constituent propositions. In this case, the propositional formula "(p ∧ q) → (p ∨ q)" is always true, because if "p ∧ q" is false, then "p ∨ q" is true, and if "p ∧ q" is true, then "(p ∧ q)" is false.

A contradiction is a propositional formula that is always false, regardless of the truth values of its constituent propositions. In this case, the propositional formula "(p → q) → (p ∧ ~q)" is always false, because if "p → q" is false, then "p ∧ ~q" is true, and if "p → q" is true, then "(p → q)" is false.

A tautology is a propositional formula that is always true, regardless of the truth values of its constituent propositions. In this case, the propositional formula "(p ∨ q) → (q ∨ p)" is always true, because if "p ∨ q" is true, then "q ∨ p" is also true, and if "p ∨ q" is false, then "(p ∨ q) → (q ∨ p)" is also true.

A contradiction is a propositional formula that is always false, regardless of the truth values of its constituent propositions. In this case, the propositional formula "(p ∧ q) → (p ∨ q)" is always false, because if "p ∧ q" is false, then "p ∨ q" is true, and if "p ∧ q" is true, then "(p ∧ q)" is false.

A tautology is a propositional formula that is always true, regardless of the truth values of its constituent propositions. In this case, the propositional formula "(p → q) → (p ∧ ~q)" is always true, because if "p → q" is false, then "p ∧ ~q" is true, and if "p → q" is true, then "(p → q)" is false.

A contradiction is a propositional formula that is always false, regardless of the truth values of its constituent propositions. In this case, the propositional formula "(p ∧ q) → (p → ~q)" is always false, because if "p ∧ q" is false, then "p → ~q" is true, and if "p ∧ q" is true, then "(p ∧ q)" is false.

A tautology is a propositional formula that is always true, regardless of the truth values of its constituent propositions. In this case, the propositional formula "(p → q) → ((q → r) → (p → r))" is always true, because if "p → q" is true, then "(q → r) → (p → r)" is also true, and if "p → q" is false, then "(p → q) → ((q → r) → (p → r))" is also true.

A contradiction is a propositional formula that is always false, regardless of the truth values of its constituent propositions. In this case, the propositional formula "((p → q) ∨ (q → r)) → (p → r)" is always false, because if "p → q" is true and "q → r" is false, then "p → r" is false, and "((p → q) ∨ (q → r)) → (p → r)" is also false.

A tautology is a propositional formula that is always true, regardless of the truth values of its constituent propositions. In this case, the propositional formula "((p ∨ q) ∧ (p → r)) → (q → r)" is always true, because if "p ∨ q" is true and "p → r" is true, then "q → r" is also true, and if "p ∨ q" is false or "p → r" is false, then "((p ∨ q) ∧ (p → r)) → (q → r)" is also true.

A contradiction is a propositional formula that is always false, regardless of the truth values of its constituent propositions. In this case, the propositional formula "((p ∧ q) ∨ (p → r)) → (q → r)" is always false, because if "p ∧ q" is true and "p → r" is false, then "q → r" is false, and "((p ∧ q) ∨ (p → r)) → (q → r)" is also false.

A tautology is a propositional formula that is always true, regardless of the truth values of its constituent propositions. In this case, the propositional formula "((p → q) ∧ (q → r)) → (p → r)" is always true, because if "p → q" is true and "q → r" is true, then "p → r" is also true, and if "p → q" is false or "q → r" is false, then "((p → q) ∧ (q → r)) → (p → r)" is also true.

A contradiction is a propositional formula that is always false, regardless of the truth values of its constituent propositions. In this case, the propositional formula "((p ∨ q) ∨ (p → r)) → (q → r)" is always false, because if "p ∨ q" is true and "p → r" is false, then "q → r" is false, and "((p ∨ q) ∨ (p → r)) → (q → r)" is also false.

A tautology is a propositional formula that is always true, regardless of the truth values of its constituent propositions. In this case, the propositional formula "((p → q) ∧ (q → r)) → (p → r)" is always true, because if "p → q" is true and "q → r" is true, then "p → r" is also true, and if "p → q" is false or "q → r" is false, then "((p → q) ∧ (q → r)) → (p → r)" is also true.