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