Logic
• Statement:
a statement is a sentence that is either true
or false
• Compound statement:
a statement that contains one or more simpler
statement
(not, and, or, if . . . then, only if, if and only if)
Determine which of the following
are statements.
1.
35  6
2. Solve the equation 2x + 5 = 3.
3. x2 + 1 = 0 has no solution.
4.
x  1  (x  1)(x  1)
5.
Is 2 a rational number?
2
Connectives
• Negation:
denial of a statement (symbol: ~)
• Conjunction:
and (symbol: ^)
• Disjunction:
or – inclusive or (symbol: V)
• Conditional:
if . . ., then (symbol: →)
Using the symbolic representations
p: The car costs $40,000.
q: The car goes 140 miles per
hour.
r: The car is red.
express the following compound
statements in symbolic form.
The car is red, goes 140 mph, and does not
cost $40,000.
Using the symbolic
representations
p: I am an environmentalist.
q: I recycle my aluminum cans.
r: I recycle my newspapers.
express the following compound
statements in symbolic form.
1. q  r   p
Negation
p
~p
T
F
F
T
Conjunction
p
q
p q
T
T
T
T
F
F
T
F
F
F
F
F
Disjunction
p
q
p q
T
T
T
T
F
T
F
T
T
F
F
F
Conditional
p
q
p q
T
T
T
T
F
F
F
T
T
F
F
T
Construct a truth table for:
p  q  ~ q
p
q
T
T
T
F
F
T
F
F
p  q  ~ q p  q  ~ q
DeMorgan’s Laws
~ p  q  ~ p  ~ q
~ p  q  ~ p  ~ q
Conditionals
Name
Conditional
Converse
Inverse
Contrapositive
Symbolic Form
p q
q p
~ p ~ q
~ q ~ p
Write the converse, inverse, and
contrapositive of the sentence:
If you do not eat meat, you are a
vegetarian
Converse: If you are a vegetarian, then
you do not eat meat.
Inverse:
If you do eat meat, then you
are not a vegetarian.
Contrapositive:
If you are not a vegetarian,
then you do eat meat.
The Dragons of Lidd
There are two types of dragons in the Kingdom of Lidd. Rational dragons, being
sensible, have determined that devouring farm animals and their owners is, in
the long run, not healthy for dragons. Predator dragons, on the other hand,
respond to their instincts and refuse to do otherwise, nor do they show any fear
of humans.
Dragons in Lidd are also of two different colors related to their veracity. Red
rational dragons always tell the truth, and red predator dragons always lie. Grey
rational dragons always tell the truth and grey predator dragons always lie.
Because dragons are few in number and are considered an endangered
species, the King has decreed that rational dragons shall be protected, and that
any knight caught slaying a rational dragon will be dealt with severely.
It would help to know which dragons are rationals and which are predators. It
would also help to know a dragon’s color. (If one catches a red dragon in a lie,
one would know that he is a predator.) Unfortunately, there is an affliction
endemic to humans in Lidd: they are colorblind. To them, all dragons look grey.
The Dragons of Lidd
One Dragon
For this dragon’s statement, construct a truth
table. Then explain how to interpret in word
each row of the table.
Dragon: I am a grey predator.
The Dragons of Lidd
Another Dragon
For this dragon’s statement, construct a truth
table. Then explain how to interpret in word
each row of the table.
Dragon: I am not a red rational.
The Dragons of Lidd
Wishy-Washy Dragons
For this dragon’s statement, construct a truth
table. Then explain how to interpret in word
each row of the table.
Dragon A: I am either red or I am rational.
The Dragons of Lidd
Forthright Dragons
For this dragon’s statement, construct a truth
table. Then explain how to interpret in word
each row of the table.
Dragon A: If I am rational, then I am red.