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TRUTH TABLES

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Introduction Statements have truth values They are either true or false but not both Statements may be simple or compound Compound statements are made up of substatements.

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Statements It is raining. The grass is wet. I did my homework. Roses are red. Violets are blue.

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Compound Statements Roses are red and violets are blue. He is very intelligent or he studies at night. My cat is hungry and he is black.

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Questions are not statements Questions cannot be true or false. –What time is it? –What color is my cat? –What grade will I get in CS230?

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TRUTH VALUE The truth or falsity of a statement is its truth value. Simple statements have a true or false truth value. –It is raining. T if it is raining F if it isn’t The truth value of a compound statement is determined by the truth value of the substatements combined with how they are connected.

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STATEMENTS Our book represents statements with the letters –p –q –r –s

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COMPOUND STATEMENT We created compound statements using connectives. –Conjunction (And) –Disjunction (Or) –Negation (Not)

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Conjunction Joining two statements with AND forms a compound statement called a conjunction. p Λ q Read as “p and q” The truth value is determined by the possible values of ITS substatements. To determine the truth value of a compound statement we create a truth table

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CONJUNCTION TRUTH TABLE pqp Λ q TTT TFF FTF FFF

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Conjunction Rule The compound statement p Λ q will only be TRUE when p is true and q is true

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Disjunction Joining two statements with OR forms a compound statement called a “disjunction. p ν q Read as “p or q” The truth value is determined by the possible values of ITS substatements. To determine the truth value of a compound statement we create a truth table

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DISJUNCTION TRUTH TABLE pqp ν q TTT TFT FTT FFF

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DISJUNCTION RULE The compound statement p ν q will only be FALSE when p is false and q is false

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NEGATION ~p read as not p Negation reverses the truth value of any statement

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NEGATION TRUTH TABLE P~P TF FT

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PROPOSITIONS AND TRUTH TABLES We can use our connectives to create compound statements that are much more complicated than just 2 substatements. When p and q become variables of a complex statement we call this a proposition. ~(pΛ~q) is an example of a proposition The truth value of a proposition depends upon the truth values of its variables so we create a truth table.

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TRUTH TABLE THE PROPOSITION ~(pΛ~q) pq~qpΛ~q~(pΛ~q) TTFFT TFTTF FTFFT FFTFT

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PROPOSITIONS AND TRUTH TABLES First Columns are always your initial variables –2 variables requires 4 rows –3 variables requires 8 rows –N variables requires 2 n rows We then create a column for each stage of the proposition and determine the truth value for the stage. The last column is the final truth value for the entire proposition.

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Creating a stepwise truth table pq~(p^~q) TTTTFFT TFFTTTF FTTFFFT FFTFFTF Step41321

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Step 1 pq~(p^~q) TTTT TFTF FTFT FFFF Step11

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Step 2 pq~(p^~q) TTTFT TFTTF FTFFT FFFTF Step121

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Step 3 pq~(p^~q) TTTFFT TFTTTF FTFFFT FFFFTF Step1321

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Step 4 pq~(p^~q) TTTTFFT TFFTTTF FTTFFFT FFTFFTF Step41321

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TAUTOLOGIES AND CONTRADICTIONS Tautology – when a proposition’s truth value (last column) consists of only T’s Contradiction – when a proposition’s truth value (last column) consists of only F’s p~pp V ~p TFT FTT p~pp Λ ~p TFF FTF

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Principle of Substitution If P(p,q,…) is a tautology then P(P 1, P 2,…) is a tautology for any propositions P 1 and P 2

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Principle of Substitution pqp^q~(p^q)(p^q) V ~(p^q) TTTFT TFFTT FTFTT FFFTT

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LOGICAL EQUIVALENCE Two propositions P(p,q,…) and Q(p,q, …) are said to be logically equivalent, or simply equivalent or equal when they have identical truth tables. ~(p Λ q) ≡ ~p V ~q

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Logical Equivalence pqp^q~(p^q) TTTF TFFT FTFT FFFT pq~p~q~pV~q TTFFF TFFTT FTTFT FFTTT

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Conditional and Biconditional Statements If p then q is a conditional statement –p q read as p implies q or p only if q P if and only if q is a biconditional statement – p q read as p if and only if q

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Conditional p q pq TTT TFF FTT FFT

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Biconditional p q pq TTT TFF FTF FFT

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Conditionals and equivalence ~p V q ≡ p q pq~p~p V q TTFT TFFF FTTT FFTT pqp q TTT TFF FTT FFT

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Converse, Inverse and Contrapositive ConditionalConverseInverseContrapositive pqp qq p~p ~q~q ~p TTTTTT TFFTTF FTTFFT FFTTTT

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Arguments An argument is a relationship between a set of propositions P 1, P 2, … called premises and another proposition Q called the conclusion. P 1, P 2, …P 8 |- Q An argument is valid if the premises yields the conclusion An argument is called a fallacy when it is not valid.

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Logical Implication A proposition P(p,q,…) is said to logically imply a proposition Q(p,q…) written P(p,q…) => Q (p,q…) if Q (p,q…) is true whenever P(p,q…) is true

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