Transcript Section 1.3 Implications

Section 1.3
Implications
Vocabulary Words
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conditional operation (⇒ or →)
conditional proposition
conditional statement (implication statement)
hypothesis (antecedent)
conclusion (consequent)
converse
contrapositive
inverse
biconditional operation (⇔ or ↔)
One more time…
• Last time I showed the following order of
precedence:
• ¬
highest
• 
• 
lowest
• And today we will add in these even lower
• , 
But…
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But, we discussed that some sources say:
~
highest
,
, 
lowest
Who is “right”
• I searched several discrete textbooks on my
bookshelf and found nearly a 50/50 split
Who is “right”
• Several programming languages agree with
your textbook (Python, Java, C++)
• But other’s disagree (Smalltalk, Ruby, and,
to some extent, Ada)
Who is “right”
Even wikipedia can’t decide…
Wikipedia - First Order Logic Wikipedia - Logical Connectives
The bottom line(s)
• I will follow your book’s rules to keep
things consistent.
• BUT, use parentheses to clear up confusion.
• AND, pay attention when you pick up a
new language
Conditional Statements
(aka Implication)
• “If I go to Fareway I will buy Diet
Mountain Dew”
• This sort of statement is known as a
conditional statement (or an implication
statement).
Conditional/Implication
• In logic this is written in the form:
pq
• And we read this as:
– If p then q
– p implies q (“Going to Fareway implies that I
will buy Diet Mountain Dew”)
Conditional/Implication
• In logic this is written in the form:
pq
• We state that p is the hypotheses or the
antecedent (or assumption or premise)
• We state that q is the conclusion or the
consequent
Conditional/Implication
• The original statement is False when p is
true and q is false; otherwise it is true.
p
q
T
T
F
F
T
F
T
F
pq
T
F
T
T
Conditional/Implication
• A conditional statement that is true by
virtue of the hypothesis being false is called
vacuously true or true by default.
• (This situation always messes with students)
p
T
q
T
pq
T
T
F
F
F
T
F
F
T
T
Logical Equivalence:
Conditional/Implication
• Notice that
( p  q)  ~p  q
p
q
pq
T
T
F
F
T
F
T
F
T
F
T
T
~p  q
T
F
T
T
In-Class Activity #1
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If you study in this course you will get an A.
Tomorrow is Friday if today is Thanksgiving.
n is prime implies n is odd or n is 2.
Tim is Ann’s father is sufficient for Jim being her uncle
and Sue being her aunt.
• n is divisible by 6 only if n is divisible by 2 and n is
divisible by 3.
• P being a rectangle is necessary for P being a square.
In-Class Activity #1
• If you study in this course you will get an A.
– Study  Get an A
• Tomorrow is Friday if today is Thanksgiving.
– Today is Thanksgiving  Tomorrow is Friday
• n is prime implies n is odd or n is 2.
– N is prime  n is odd or n is 2
In-Class Activity #1
• Tim is Ann’s father is sufficient for Jim being her
uncle and Sue being her aunt.
– Tim is Ann’s father  Jim is her uncle and
Sue is her aunt
• n is divisible by 6 only if n is divisible by 2 and n
is divisible by 3.
– N is divisible by 6  n is divisible by 2 and n
is divisible by 3
• P being a rectangle is necessary for P being a
square.
– P is a square  P is a rectangle
Order of Precedence
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¬


, 
highest
lowest
In-Class Activity #2
p
q
¬ p  q  ¬q
T
T
(1)
T
F
(2)
F
T
(3)
F
F
(4)
What is the negation of an
implication statement
• How do we write:
¬( p  q)
• Remember that:
( p  q)  ¬ p  q
• Therefore:
¬( p  q)  ¬ p  q)  p  ¬ q
Negation of an implication
• ¬( p  q)  p  ¬ q
• What is the negation of “If I go to Fareway
I will buy Diet Mountain Dew”
• I will go to Fareway but I will NOT buy
Diet Mountain Dew
• Notice that this does NOT start with an if.
Contrapositive of an Implication
• The contrapositive of the conditional
statement “if p then q” is written as “if not q
then not p”
• The contrapositive of p  q is written as
¬q¬p
So what…
• Is a conditional statement logically
equivalent to it’s contrapositive?
• Build the truth table and you will see that it
is!
The Converse of an Implication
• The converse of the conditional statement
“if p then q” is written as “if q then p”
• The converse of p  q is written as
qp
So what…
• Is a conditional statement logically
equivalent to it’s converse?
• Build the truth table and you will see that it
is not!
• However, this is one of the most common
logic errors made by beginning students.
The Inverse of an Implication
• The inverse of the conditional statement “if
p then q” is written as “if not p then not q”
• The inverse of p  q is written as
¬p¬q
So what…
• Is a conditional statement logically
equivalent to it’s inverse?
• Build the truth table and you will see that it
is not!
• HOWEVER, the converse and the inverse
are logically equivalent.
– One is the contrapositive of the other
In-Class Activity #3
• If my client is guilty, then the knife was in
the drawer
• Write the:
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Converse
Contrapositive
Inverse
Negation
Converse
• qp
• If the knife was in the drawer then my client
is guilty.
Contrapositive
• ¬q¬p
• If the knife was not in the drawer then my
client is not guilty.
Inverse
• ¬p¬q
• If my client is not guilty than the knife is
not in the drawer.
Negation
• p^¬q
• My client is guilty but the knife is not in the
drawer.
• Notice that the negation of implication is
NOT an implication.