Propositions and Truth Tables

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 = 23NOYESNONOYES

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)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: ANDLet 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.

Disjunction of p and q: ORLet 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.

Two types of ORINCLUSIVE OR means either or both

EXCLUSIVE OR means one or the other, but not both

Two types of Disjunction of p and q: OR

ImplicationsIf p, then qp implies qif p, qp only if qp is sufficient for qq if pq whenever pq is necessary for pProposition p = antecedent

Proposition q = consequent

Converse, Inverse, ContrapositiveConditional p qContrapositive of p q is the proposition q p .Converse of p q is q pInverse of p q is p qIf you are not breathing, then you are not sleepingp You are sleeping q you are breathingIf you are sleeping, then you are breathing.If you are breathing, then you are sleeping.If you are not sleeping, then you are not breathing.

Find the conditional, converse, inverse and contrapositive:Conditional p qContrapositive of p q is the proposition q p Converse of p q is q pInverse of p q is p qIf the sun is shining, then it is warm outside.p The sun is shinning q it is warm outsideIf it is warm outside, then the sun is shining.If the sun is not shining, then it is not warm outside.If it is not warm outside,the sun is not shining.

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

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.

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)

(T/F) Conditional

(T/F) Converse

(T/F)Inverse

(T/F)Contrapositive

p mC = 100 q ABC is obtuse If mC = 100, then ABC is obtuse.If ABC is obtuse then mC = 100, If mC 100, then ABC is not obtuse.If ABC is not obtuse, then mC 100

(T/F) Conditional

(T/F) Converse

(T/F)Inverse

(T/F)Contrapositive

If ABC is isosceles, then it is equilateralIf ABC is not equilateral, then it is not isoscelesIf ABC is not isosceles, then it is not equilateral If ABC is equilateral, then it is isoscelesp ABC is equilateral q it is isosceles

(T/F) Conditional

(T/F) Converse

(T/F)Inverse

(T/F)Contrapositive

p G is the midpoint of KL q GQ bisects KL

Let R = I work at this school and D = My name is Ms. D

Translate the following symbols into sentences, and indicate T/F:

D R: ___ (T/F)

b. R D: ___ (T/F)

c. ~D ~R: ___ (T/F)

If my name is Ms. D, then I work at this school.If I work at this school, then my name is Ms. D.If my name is not Ms. D, then I do notwork at this school.

5. Given:a. If today is warm, the pool will be crowded.b. If it rains today, the pool will not be crowded.c. Either today is warm or I will wear a long-sleeved shirt.d. It will not rain today.Using W, P, R, S, & proper connectives (~, , etc.), express each sentence into symbolic form.Let W represent Today is warm.Let P represent The pool will be crowded.Let R represent It rains today.Let S represent I will wear a long-sleeved shirt.

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