Transcript Wrapping up Section 1.3 Section 1.4 Biconditionals Logical Equivalence

Wrapping up Section 1.3
Section 1.4
Biconditionals
Logical Equivalence
In-Class Activity #1
• Complete the following Truth Table
p
q
T
T
T
F
F
T
F
F
¬p
¬q
Original
A
B
C
D
pq
qp
¬q¬p
¬p¬q
p^¬q
In-Class Activity #1
• Complete the following Truth Table
Original
A
B
C
D
p
q
¬p
¬q
pq
qp
¬q¬p
¬p¬q
p^¬q
T
T
F
F
T
T
T
T
F
T
F
F
T
F
T
F
T
T
F
T
T
F
T
F
T
F
F
F
F
T
T
T
T
T
T
F
In-Class Activity #2
• Assuming that p  q is the original implication, label
columns A-D with their proper vocabulary names.
Original
A
B
C
D
p
q
¬p
¬q
pq
qp
¬q¬p
¬p¬q
p^¬q
T
T
F
F
T
T
T
T
F
T
F
F
T
F
T
F
T
T
F
T
T
F
T
F
T
F
F
F
F
T
T
T
T
T
T
F
In-Class Activity #2
• Assuming that p  q is the original implication, label
columns A-D with their proper vocabulary names.
Original
Inverse Contrapositive
Converse
Negation
p
q
¬p
¬q
pq
qp
¬q¬p
¬p¬q
p^¬q
T
T
F
F
T
T
T
T
F
T
F
F
T
F
T
F
T
T
F
T
T
F
T
F
T
F
F
F
F
T
T
T
T
T
T
F
Consider
• Last time I made the statement:
– If I go to Fareway I will buy Diet Mountain
Dew.
p = “I go to Fareway”
q = “I will buy DMD”
• And we learned we could write this with an
implication:
If p then q
pq
Consider
• We also learned that this can be written as:
– I will buy DMD if I go to Fareway.
• Which is:
q if p
• But still written as
pq
Consider
• How is this statement different if instead I
make the statement:
– I will buy DMD only if I go to Fareway.
• Notice that this is:
– q only if p
Consider
• I will buy DMD only if I go to Fareway.
– q only if p
• Notice that this is the same as saying:
– If I don’t go to Fareway then I will not buy
DMD
¬p¬q
– Whose contrapositive is:
qp
So Let’s Summarize
• I will buy DMD if I go to Fareway.
pq
• I will buy DMD only if I go to Fareway.
qp
• So what about:
– I will buy DMD if and only if I go to Fareway:
pqqp
which we instead write as: p  q
Biconditional
• The Biconditional of p and q is written “p, if and
only if, q”
• It is denoted p  q
• It is true if both p and q have the same truth values and
is false otherwise.
pq
p
q
T
T
T
T
F
F
F
T
F
F
F
T
Biconditional
• The Biconditional of p and q is written “p, if and
only if, q”
• It is denoted p  q
• It is true if both p and q have the same truth values and
is false otherwise. WHY
p
q
pq
qp
(p  q)  (q  p)
pq
T
T
T
F
F
T
F
F
Vocabulary Words
• Tautology
• Contradiction
• Logically Equivalent
In-Class Activity #3
Define these three terms
• Tautology
• Contradiction
• Logically Equivalent
In-Class Activity #3
Define these three terms
• Tautology – A compound proposition that is
always true
• Contradiction – A compound proposition that is
never true
• Logically Equivalent – Two compound
propositions that result in the same truth value
regardless of the truth values of their individual
propositions.
Logically Equivalent
• Which of the expressions from our earlier activity are
logically equivalent
Original
Inverse Contrapositive
Converse
Negation
p
q
¬p
¬q
pq
qp
¬q¬p
¬p¬q
p^¬q
T
T
F
F
T
T
T
T
F
T
F
F
T
F
T
F
T
T
F
T
T
F
T
F
T
F
F
F
F
T
T
T
T
T
T
F
Laws of Propositional Logic
(Logically Equivalent Statements)
Using truth tables, confirm the
following laws