Conditional statement or implication IF p then q is denoted p ⇒ q p is the antecedent or hypothesis q is the consequent or conclusion ⇒ means IF…THEN To form the implication p: I am hungry q: I will eat p: It is snowing q: = 8 IF I am hungry, THEN I will eat IF it is snowing, THEN 3+5=8 Normally you would use statements that have cause and effect.

IF p then q (p ⇒ q) is only considered false if p is true and q is false. p = t, q = f results in F pq p ⇒ q TTT TFF FTT FFT

IF p then q: p ⇒ q means all of the following: p implies q q if p p only if q p is a sufficient condition for q q is a necessary condition for p Converse of p ⇒ q is the implication q ⇒ p (IF q then p) Contrapositive of p ⇒ q is the implication q ⇒ p (if not q then not p)

IF it is raining, THEN I get wet p: It is raining q: I get wet Converse q ⇒ p (if q then p) IF I get wet, THEN it is raining Contrapositive q ⇒ p (if not q then not p) IF I do not get wet, then it is not raining

Equivalence or bi conditional ⇔ IF and only IF p ⇔ q (p if and only if q) is only true when both p and q are true or when both p and q are false pq p ⇔ q TTT TFF FTF FFT

3 >2 IF and only IF 0 < 3-2 p: 3 >2 q: 0 < 3-2 p ⇔ q (p if and only if q) is true because p is true and q is true.

A statement that is true for all possible values of its propositional variables is called a tautology. In other words, if the result column only has T’s in it, the statement is a tautology. Review Tautology handout Contradiction or Absurdity: is a statement that is always fasle: p ∧ p (p and not p) will always have a result of false. Contingency: a statement that can either be true or false, depending on the truth values of its propositional variables. (p ⇒ q) ∧ (p ∨ q)

Logically equivalent is denoted by ≡ IF p ⇔ q is a tautology then the two propositions are equal. Page 58 correction Last paragraph in example 6 the second & third should be third & fourth Review Page 59 properties Page 60: ∀ means for all ∃ means there exists