Transcript TRUTH TABLES

TRUTH TABLES
Introduction
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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
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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
pΛq
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
pνq
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
4
1
3
2
1
Step
Step 1
p
q
T
T
T
T
T
F
T
F
F
T
F
T
F
F
F
F
1
1
Step
~
(p
^
~
q)
Step 2
p
q
T
T
T
~
q)
T
F
T
F
T
T
F
F
T
F
F
T
F
F
F
T
F
1
2
1
Step
~
(p
^
Step 3
p
q
T
(p
^
~
q)
T
T
F
F
T
T
F
T
T
T
F
F
T
F
F
F
T
F
F
F
F
T
F
1
3
2
1
Step
~
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
4
1
3
2
1
Step
TAUTOLOGIES AND
CONTRADICTIONS
• Tautology – when a
proposition’s truth
value (last column)
consists of only T’s
p
~p
p V ~p
T
F
T
F
T
T
• Contradiction – when
a proposition’s truth
value (last column)
consists of only F’s
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)
p
q ~p ~q ~pV~q
T T T
F
T
T F
F
F
T F F
T
T
F F
T
T
F T F
T
F
T T
F
T
F F 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
p
q
pq
T
T
F
T
T
T
T
T
F
F
F
T
F
F
F
T
T
T
F
T
T
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