## Equivalence

Equivalent statement forms always have the same truth value, and we assert that fact by expressing the equivalence in the form of a biconditional. Such statements have the same truth value on every line of the corresponding truth table.

The importance of this point for our purposes is that equivalent statements can be substituted for each other in a proof.

 Tautology (Taut) Double Negation (DN) p (p p) p ~ ~ p (p v p)
 Commutation (Com) Association (Assoc) (p q) (q p) [p (q r)] [(p q) r] (p v q) (q v p) [p v (q v r)] [(p v q) v r]
 Distribution (Dist) De Morgan's Law (DM) [p (q v r)] [(p q) v (p r)] ~ (p q) (~ p v ~ q) [p v (q r)] [(p v q) (p v r)] ~ (p v q) (~ p ~ q)
 Contraposition (Contra) Implication (Imp) (p q) (~ q ~ p) (p q) (~ p v q) ~ (p ~ q)
 Biconditional (Bicon) Exportation (Exp) (p q) [(p q) (q p)] [(p q) r] [p (q r)] [(p q) v (~ p ~ q)]

Comprehension Questions

 1. Identify the following equivalence: p (p v p). a) tautology b) commutation c) distribution d) contraposition e) double negation 2. Identify the following equivalence: [(p . q) > r] [p > (q > r)]. a) association b) De Morgan's law c) implication d) biconditional e) exportation 3. Identify the following equivalence: ~(p . q) ( ~p v ~q). a) association b) De Morgan's law c) implication d) biconditional e) exportation 4. Identify the following equivalence: [p v (q v r)] [(p v q) v r]. a) association b) De Morgan's law c) implication d) biconditional e) exportation 5. Identify the following equivalence: (p v q) (q v p). a) tautology b) commutation c) distribution d) contraposition e) double negation 6. Identify the following equivalence: (p > q) (~q > ~p). a) tautology b) commutation c) distribution d) contraposition e) double negation 7. Identify the following equivalence: p ~~p. a) tautology b) commutation c) distribution d) contraposition e) double negation 8. Identify the following equivalence: (p > q) (~p v q). a) association b) De Morgan's law c) implication d) biconditional e) exportation 9. Identify the following equivalence: [p . (q v r)] [(p . q) v (p . r)]. a) tautology b) commutation c) distribution d) contraposition e) double negation 10. Identify the following equivalence: (p . q) (q . p). a) tautology b) commutation c) distribution d) contraposition e) double negation 11. Identify the following equivalence: ~(p v q) (~p . ~q). a) association b) De Morgan's law c) implication d) biconditional e) exportation 12. Identify the following equivalence: (p = q) [(p > q).(q > p)]. a) association b) De Morgan's law c) implication d) biconditional e) exportation 13. Identify the following equivalence: [p.(q.r)] [(p.q).r]. a) association b) De Morgan's law c) implication d) biconditional e) exportation 14. Identify the following equivalence: [p v (q . r)] [(p v q).(p v r)] . a) tautology b) commutation c) distribution d) contraposition e) double negation 15. Identify the following equivalence: (p = q) [(p . q) v (~p . ~q)]. a) association b) De Morgan's law c) implication d) biconditional e) exportation 16. Identify the following equivalence: (p > q) ~(p .~ q) . a) association b) De Morgan's law c) implication d) biconditional e) exportation

Conditional Proof and Reductio ad Absurdum