Propositions and Truth Tables

Proposition: Makes a claim that may be either true or false; it must have the structure of a complete sentence.

Are these propositions? Over the mountain and through the woods. All apples are fruit. The quick, brown fox. Are you here? 2 + 3 = 23 NO YES NO NO YES

Negation of p Let p be a proposition. The statement “It is not the case that p” is also a proposition, called the “negation of p” or p (read “not p”) Table 1. The Truth Table for the Negation of a Proposition p p T F F T p = The sky is blue. p = It is not the case that the sky is blue. p = The sky is not blue.

Conjunction of p and q: AND Let p and q be propositions. The proposition “p and q,” denoted by pq is true when both p and q are true and is false otherwise. This is called the conjunction of p and q. Table 2. The Truth Table for the Conjunction of two propositions p q pq T T T T F F F T F F F F

Disjunction of p and q: OR Let p and q be propositions. The proposition “p or q,” denoted by pq, is the proposition that is false when p and q are both false and true otherwise. Table 3. The Truth Table for the Disjunction of two propositions p q pq T T T T F T F T T F F F

Two types of OR INCLUSIVE OR means “either or both” EXCLUSIVE OR means “one or the other, but not both”

Two types of Disjunction of p and q: OR INCLUSIVE OR means “either or both” p q pq T T T T F T F T T F F F EXCLUSIVE OR means “one or the other, but not both” p q pq T T F T F T F T T F F F

Implications If p, then q p implies q if p, q p only if q p is sufficient for q q if p q whenever p q is necessary for p Proposition p = antecedent Proposition q = consequent

Converse, Inverse, Contrapositive Conditional p  q If you are sleeping, then you are breathing. Converse of p  q is q  p If you are breathing, then you are sleeping. Inverse of p  q is p  q If you are not sleeping, then you are not breathing. Contrapositive of p  q is the proposition  q  p If you are not breathing, then you are not sleeping.

Find the converse, inverse and contrapositive: Conditional p  q If the sun is shining, then it is warm outside. Converse of p  q is q  p Inverse of p  q is p  q Contrapositive of p  q is the proposition  q  p

Biconditional Let p and q be propositions. The biconditional pq is the proposition that is true when p and q have the same truth values and is false otherwise. “p if and only if q, p is necessary and sufficient for q” Table 6. The Truth Table for the biconditional pq. p q pq T T T T F F F T F F F T

Logical Equivalence An important technique in proofs is to replace a statement with another statement that is “logically equivalent.” Tautology: compound proposition that is always true regardless of the truth values of the propositions in it. Contradiction: Compound proposition that is always false regardless of the truth values of the propositions in it.

Logically Equivalent Compound propositions P and Q are logically equivalent if PQ is a tautology. In other words, P and Q have the same truth values for all combinations of truth values of simple propositions. This is denoted: PQ (or by P Q)