Section 6.3 Lecture
These are the truth tables we have covered so far:
Negation (NOT)
p
~p
T
F
F
T
“change to opposite”
Two more:
Implication (If…then)
p
q
p→q
T
T
T
F
F
T
F
F
Conjunction (AND)
p
q
pᴧq
T
T
T
T
F
F
F
T
F
F
F
F
“only T if both are T”
Disjunction (OR)
p
q
pVq
T
T
T
T
F
T
F
T
T
F
F
F
“only F if both are F”
Biconditional (If and only if)
p
q
p↔q
T
T
T
F
F
T
F
F
Example: True or False: (3=7-5) → (1>6)
Example: Write in symbolic form:
“You will like physics if you understand math or I am a frog.”
(Let p=”You will like physics.” and q=”You understand math.” and r=”I am a frog.”)
Example: Find the truth value of ~q→r.
Assume q is true and r is true.
Implication (If…then)
p
q
p→q
T
T
T
T
F
F
F
T
T
F
F
T
Negation (NOT)
p
~p
T
F
F
T
“change to opposite”
Conjunction (AND)
p
q
pᴧq
T
T
T
T
F
F
F
T
F
F
F
F
“only T if both are T”
Implication (If…then)
p
q
p→q
T
T
T
T
F
F
F
T
T
F
F
T
“only F when p is T & q is F”
Disjunction (OR)
p
q
pVq
T
T
T
T
F
T
F
T
T
F
F
F
“only F if both are F”
Biconditional (If and only if)
p
q
p↔q
T
T
T
T
F
F
F
T
F
F
F
T
“only T if p & q match”
Example: Fill in the blanks:
p
T
T
F
F
q
T
F
T
F
~p
~pᴧq
q→p
(~pᴧq) ᴧ (q→p)
Negation (NOT)
p
~p
T
F
F
T
“change to opposite”
Conjunction (AND)
p
q
pᴧq
T
T
T
T
F
F
F
T
F
F
F
F
“only T if both are T”
Two more:
Implication (If…then)
p
q
p→q
T
T
T
T
F
F
F
T
T
F
F
T
“only F when p is T & q is F”
Biconditional (If and only if)
p
q
p↔q
T
T
T
T
F
F
F
T
F
F
F
T
“only T if p & q match”
Example: Complete the truth table.
p q r (~r→q)V(~p→r)
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
Disjunction (OR)
p
q
pVq
T
T
T
T
F
T
F
T
T
F
F
F
“only F if both are F”
(You already know these)
De Morgan’s Laws
~(p V q) ≡ ~p Λ ~q
~(p Λ q) ≡ ~p V ~q
(Two more to learn)
Laws Involving Implications
p → q ≡ ~p V q
~(p → q) ≡ p Λ ~q
Example: Write
“If I am tired, then I can make coffee.”
without using the “If…then” connective.
Example: Write the negation of
“If I am tired, then I can make coffee.”
without using the “If…then” connective.
Example: Use truth tables to determine if
~q→~r is equivalent to ~r V q.
q
T
T
F
F
r ~q→~r
T
F
T
F
Implication (If…then)
p
q
p→q
T
T
T
F
F
T
F
F
q
T
T
F
F
r ~r V q
T
F
T
F
Basic circuit
(with gate that can open & close)
Circuit "in series"
-Gates p and q are lined up
one after the other.
-'Current' will only flow all the way
across if both gates are closed.
-If 'open' is False & 'closed' is True,
this circuit is logically equivalent to the .
CONJUNCTION ( p Λ q )
Circuit "in parallel"
-Gates p and q are arranged
with one above the other.
-'Current' will not flow all the way
across only if both gates are open.
-If we define 'open' as False &
'closed' as True, this circuit is logically
equivalent to the
DISJUNCTION ( p V q ).
More complex circuits can be built using simple circuits.
Example:
The statement: [ ( p Λ q ) V q ] Λ r is
logically equivalent to the circuit to the
right.