truth tables. introduction statements have truth values they are either true or false but not both...

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

p q p Λ q

T T T

T F F

F T F

F F F

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

p q p ν q

T T T

T F T

F T T

F F F

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

T F

F T

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)

p q ~q pΛ~q ~(pΛ~q)

T T F F T

T F T T F

F T F F T

F F T F T

PROPOSITIONS AND TRUTH TABLES

• First Columns are always your initial variables– 2 variables requires 4 rows– 3 variables requires 8 rows– N variables requires 2n 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

p q ~ (p ^ ~ q)

T T T T F F T

T F F T T T F

F T T F F F T

F F T F F T F

Step 4 1 3 2 1

Step 1

p q ~ (p ^ ~ q)

T T T T

T F T F

F T F T

F F F F

Step 1 1

Step 2

p q ~ (p ^ ~ q)

T T T F T

T F T T F

F T F F T

F F F T F

Step 1 2 1

Step 3

p q ~ (p ^ ~ q)

T T T F F T

T F T T T F

F T F F F T

F F F F T F

Step 1 3 2 1

Step 4

p q ~ (p ^ ~ q)

T T T T F F T

T F F T T T F

F T T F F F T

F F T F F T F

Step 4 1 3 2 1

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 ~p p V ~p

T F T

F T T

p ~p p Λ ~p

T F F

F T F

Principle of Substitution

• If P(p,q,…) is a tautology then P(P1, P2,…) is a tautology for any propositions P1 and P2

Principle of Substitution

p q p^q ~(p^q) (p^q) V ~(p^q)

T T T F T

T F F T T

F T F T T

F F F T T

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

p q p^q ~(p^q)

T T T F

T F F T

F T F T

F F F T

p q ~p ~q ~pV~q

T T F F F

T F F T T

F T T F T

F F T T T

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 p q p q

T T T

T F F

F T T

F F T

Biconditional

• p q p q p q

T T T

T F F

F T F

F F T

Conditionals and equivalence~p V q ≡ p q

p q ~p ~p V q

T T F T

T F F F

F T T T

F F T T

p q p q

T T T

T F F

F T T

F F T

Converse, Inverse and Contrapositive

Conditional Converse Inverse Contrapositive

p q p q q p ~p ~q ~q ~p

T T T T T T

T F F T T F

F T T F F T

F F T T T T

Arguments

• An argument is a relationship between a set of propositions P1, P2, … called premises and another proposition Q called the conclusion.

• P1, P2, …P8 |- 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