TRUTH TABLES

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.

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

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.

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?

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.

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

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

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

CONJUNCTION TRUTH TABLE pqp Λ q TTT TFF FTF FFF

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

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

DISJUNCTION TRUTH TABLE pqp ν q TTT TFT FTT FFF

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

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

NEGATION TRUTH TABLE P~P TF FT

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.

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

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.

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

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

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

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

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

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

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

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

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

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

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

Conditional p  q pq TTT TFF FTT FFT

Biconditional p  q pq TTT TFF FTF FFT

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

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

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.

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