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**Propositions and Truth Tables**

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

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**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

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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.

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**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 propositions p q pq T T T T F F F T F F F F

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**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 propositions p q pq T T T T F T F T T F F F

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**Two types of OR INCLUSIVE OR means “either or both”**

EXCLUSIVE OR means “one or the other, but not both”

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**Two types of Disjunction of p and q: OR**

INCLUSIVE OR means “either or both” p q pq T T T T F T F T T F F F EXCLUSIVE OR means “one or the other, but not both” p q pq T T F T F T F T T F F F

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**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

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**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.

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**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

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Biconditional Let 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 pq T T T T F F F T F F F T

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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.

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Logically Equivalent Compound 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)

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