2 Proposition: Makes a claim that may be either true or false; it must have the structure of a complete sentence.
3 Are these propositions? Over the mountain and through the woods. All apples are fruit.The quick, brown fox.Are you here?2 + 3 = 23NOYESNONOYES
4 Negation of pLet 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 theNegation of a Propositionp pT FF Tp = The sky is blue.p = It is not the case that the sky is blue.p = The sky is not blue.
5 Conjunction of p and q: AND Let p and q be propositions. The proposition “p and q,” denoted by pq 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 propositionsp q pqT T TT F FF T FF F F
6 Disjunction of p and q: OR Let p and q be propositions. The proposition “p or q,” denoted by pq, 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 propositionsp q pqT T TT F TF T TF F F
7 Two types of OR INCLUSIVE OR means “either or both” EXCLUSIVE OR means “one or the other, but not both”
8 Two types of Disjunction of p and q: OR INCLUSIVE OR means “either or both”p q pqT T TT F TF T TF F FEXCLUSIVE OR means “one or the other, but not both”p q pqT T FT F TF T TF F F
9 Implications If p, then q p implies q if p, q p only if q p is sufficient for qq if pq whenever pq is necessary for pProposition p = antecedentProposition q = consequent
10 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.
11 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 pInverse of p q is p qContrapositive of p q is the proposition q p
12 BiconditionalLet p and q be propositions. The biconditional pq 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 pq.p q pqT T TT F FF T FF F T
13 Logical EquivalenceAn 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.
14 Logically EquivalentCompound propositions P and Q are logically equivalent if PQ 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: PQ (or by P Q)