An old farmer used a horse to till his fields' One day, the horse ran away, and when the farmers nei

1
An ancient Chinese story
An old farmer used a horse to till his fields.
One day, the horse ran away, and when the
farmers neighbors sympathized with the old man
over his bad luck, the farmer shrugged his
shoulders and replied, Bad luck? Good luck? Who
knows?
2
An ancient Chinese story
An old farmer used a horse to till his fields.
One day, the horse ran away, and when the
farmers neighbors sympathized with the old man
over his bad luck, the farmer shrugged his
shoulders and replied, Bad luck? Good luck? Who
knows? A week later, the horse returned with a
herd of wild mares, and this time the neighbors
congratulated the farmer on his good luck. His
reply was, Good luck? Bad luck? Who knows?
3
An ancient Chinese story
An old farmer used a horse to till his fields.
One day, the horse ran away, and when the
farmers neighbors sympathized with the old man
over his bad luck, the farmer shrugged his
shoulders and replied, Bad luck? Good luck? Who
knows? A week later, the horse returned with a
herd of wild mares, and this time the neighbors
congratulated the farmer on his good luck. His
reply was, Good luck? Bad luck? Who
knows? Then, when the farmers son was
attempting to tame one of the wild horses, he
fell and broke his leg. Everyone agreed this was
very bad luck. But the farmers only reaction
was, Bad luck? Good luck? Who knows?
4
An ancient Chinese story
An old farmer used a horse to till his fields.
One day, the horse ran away, and when the
farmers neighbors sympathized with the old man
over his bad luck, the farmer shrugged his
shoulders and replied, Bad luck? Good luck? Who
knows? A week later, the horse returned with a
herd of wild mares, and this time the neighbors
congratulated the farmer on his good luck. His
reply was, Good luck? Bad luck? Who
knows? Then, when the farmers son was
attempting to tame one of the wild horses, he
fell and broke his leg. Everyone agreed this was
very bad luck. But the farmers only reaction
was, Bad luck? Good luck? Who knows? A week
later, the army marched into the village
and drafted all the young men they could find.
When they saw the farmers son with his
broken leg, they let him stay behind. Good
luck? Bad luck?
5
An ancient Chinese story
An old farmer used a horse to till his fields.
One day, the horse ran away, and when the
farmers neighbors sympathized with the old man
over his bad luck, the farmer shrugged his
shoulders and replied, Bad luck? Good luck? Who
knows? A week later, the horse returned with a
herd of wild mares, and this time the neighbors
congratulated the farmer on his good luck. His
reply was, Good luck? Bad luck? Who
knows? Then, when the farmers son was
attempting to tame one of the wild horses, he
fell and broke his leg. Everyone agreed this was
very bad luck. But the farmers only reaction
was, Bad luck? Good luck? Who knows? A week
later, the army marched into the village
and drafted all the young men they could find.
When they saw the farmers son with his
broken leg, they let him stay behind. Good
luck? Bad luck? As you can see we never know
why things work out the way they do. Try not to
blame nor complain, but rather look within
yourself. Do not lose hope nor determination,
but pursue your dreams. Always do the best that
you can and, though at times it is not easy,
believe that what has happened is for the best
and work with what is given to you.
6
An ancient Chinese story
An old farmer used a horse to till his fields.
One day, the horse ran away, and when the
farmers neighbors sympathized with the old man
over his bad luck, the farmer shrugged his
shoulders and replied, Bad luck? Good luck? Who
knows? A week later, the horse returned with a
herd of wild mares, and this time the neighbors
congratulated the farmer on his good luck. His
reply was, Good luck? Bad luck? Who
knows? Then, when the farmers son was
attempting to tame one of the wild horses, he
fell and broke his leg. Everyone agreed this was
very bad luck. But the farmers only reaction
was, Bad luck? Good luck? Who knows? A week
later, the army marched into the village
and drafted all the young men they could find.
When they saw the farmers son with his
broken leg, they let him stay behind. Good
luck? Bad luck? As you can see we never know
why things work out the way they do. Try not to
blame nor complain, but rather look within
yourself. Do not lose hope nor determination,
but pursue your dreams. Always do the best that
you can and, though at times it is not easy,
believe that what has happened is for the best
and work with what is given to you.
7
Sec. 3.1 Statements and Logical Connectives
8
Sec. 3.1 Statements and Logical Connectives
The Greeks were the first to systematically
analyze the process of human thought and arrive
at conclusions.
9
Sec. 3.1 Statements and Logical Connectives
The Greeks were the first to systematically
analyze the process of human thought and arrive
at conclusions.
Aristotle (384-322 B.C.) organized the study of
logic in his work Organon. As a result of his
work Aristotle is referred to as the father of
logic, and Aristotelian logic has been taught and
studied from this period in Aristotles life to
date.
10
Sec. 3.1 Statements and Logical Connectives
Gottfried Wilhelm Leibniz (1646-1716) had a deep
conviction that all mathematical and scientific
concepts could be derived from logic and became
the first serious student of Symbolic Logic.
11
Sec. 3.1 Statements and Logical Connectives
Gottfried Wilhelm Leibniz (1646-1716) had a deep
conviction that all mathematical and scientific
concepts could be derived from logic and became
the first serious student of Symbolic Logic.
George Boole (1815-1864), who was a self educated
English Mathematician, is considered to be the
founder of symbolic logic because of his
impressive work in this area.
12
Sec. 3.1 Statements and Logical Connectives
George Boole, Augustus De Morgan, and other
mathematicians of the nineteenth century were
anxious to make logic an abstract science that
would operate like algebra but be applicable to
all fields. One of the problems logicians faced
was that verbal language could be ambiguous and
could easily lead to confusion and contradiction.
13
Sec. 3.1 Statements and Logical Connectives
George Boole, Augustus De Morgan, and other
mathematicians of the nineteenth century were
anxious to make logic an abstract science that
would operate like algebra but be applicable to
all fields. One of the problems logicians faced
was that verbal language could be ambiguous and
could easily lead to confusion and contradiction.

The mathematician Charles Dodgson (better known
as Lewis Carroll) incorporated many interesting
ideas from logic into his books Alices
Adventures in Wonderland and Through the Looking
Glass along with other of his childrens stories.

14
Sec. 3.1 Statements and Logical Connectives
Comedians Bud Abbott and Lou Costello had fun
with the ambiguity of language in their skit
about the baseball players Whos on first,
Whats on second, I Dont know is on third
Yeah, but whos on first? (in the film The
Naughty Nineties, 1945).
http//www.youtube.com/watch?vsShMA85pv8M
15
Sec. 3.1 Statements and Logical Connectives
In logic, a statement is a sentence that can be
assigned a truth value (either true or false).
16
Sec. 3.1 Statements and Logical Connectives
In logic, a statement is a sentence that can be
assigned a truth value (either true or false).
ex. 1. The earth is flat
17
Sec. 3.1 Statements and Logical Connectives
In logic, a statement is a sentence that can be
assigned a truth value (either true or false).
ex. 1. The earth is flat 2. The door is
closed
18
Sec. 3.1 Statements and Logical Connectives
In logic, a statement is a sentence that can be
assigned a truth value (either true or false).
ex. 1. The earth is flat 2. The door is
closed 3. Bush is President of the United
States
19
Sec. 3.1 Statements and Logical Connectives
In logic, a statement is a sentence that can be
assigned a truth value (either true or false).
ex. 1. The earth is flat 2. The door is
closed 3. Bush is President of the United
States
Sentences which are NOT statements
20
Sec. 3.1 Statements and Logical Connectives
In logic, a statement is a sentence that can be
assigned a truth value (either true or false).
ex. 1. The earth is flat 2. The door is
closed 3. Bush is President of the United
States
Sentences which are NOT statements 1. Close
the door
21
Sec. 3.1 Statements and Logical Connectives
In logic, a statement is a sentence that can be
assigned a truth value (either true or false).
ex. 1. The earth is flat 2. The door is
closed 3. Bush is President of the United
States
Sentences which are NOT statements 1. Close
the door 2. Is this class interesting?
22
Sec. 3.1 Statements and Logical Connectives
"And" statements Conjunction ("and",
"but", however", nevertheless"), symbol
23
Sec. 3.1 Statements and Logical Connectives
"And" statements Conjunction ("and",
"but", however", nevertheless"), symbol
Let p Paul is a student q Paul has a car
24
Sec. 3.1 Statements and Logical Connectives
"And" statements Conjunction ("and",
"but", however", nevertheless"), symbol
Let p Paul is a student q Paul has a car
ex. Write the following statements in symbolic
form a. Paul is a student and
Paul has a car b. Paul has a
car and Paul is a student
25
Sec. 3.1 Statements and Logical Connectives
"And" statements Conjunction ("and",
"but", however", nevertheless"), symbol
Let p Paul is a student q Paul has a car
ex. Write the following statements in symbolic
form a. Paul is a student and
Paul has a car
p q b. Paul has a car
and Paul is a student

26
Sec. 3.1 Statements and Logical Connectives
"And" statements Conjunction ("and",
"but", however", nevertheless"), symbol
Let p Paul is a student q Paul has a car
ex. Write the following statements in symbolic
form a. Paul is a student and
Paul has a car
p q b. Paul has a car
and Paul is a student
q p
27
Sec. 3.1 Statements and Logical Connectives
"Not" statements Negation, symbol
28
Sec. 3.1 Statements and Logical Connectives
"Not" statements Negation, symbol
Let r Today is Thursday s It is snowing
29
Sec. 3.1 Statements and Logical Connectives
"Not" statements Negation, symbol
Let r Today is Thursday s It is snowing
ex. Write the following statements in symbolic
form a. Today is Thursday and it
is not snowing.
b. Today is not Thursday
and it is snowing.

30
Sec. 3.1 Statements and Logical Connectives
"Not" statements Negation, symbol
Let r Today is Thursday s It is snowing
ex. Write the following statements in symbolic
form a. Today is Thursday and it
is not snowing.
r s b. Today is not
Thursday and it is snowing.

31
Sec. 3.1 Statements and Logical Connectives
"Not" statements Negation, symbol
Let r Today is Thursday s It is snowing
ex. Write the following statements in symbolic
form a. Today is Thursday and it
is not snowing.
r s b. Today is not
Thursday and it is snowing.
r s
32
Sec. 3.1 Statements and Logical Connectives
Or" statements Disjunction, symbol
33
Sec. 3.1 Statements and Logical Connectives
Or" statements Disjunction, symbol
Let p Mary will go to the movie q Mary
will eat popcorn
34
Sec. 3.1 Statements and Logical Connectives
Or" statements Disjunction, symbol
Let p Mary will go to the movie q Mary
will eat popcorn
ex. Write the following statements in symbolic
form   a. Mary will go to the movie or Mary will
eat popcorn
b. Mary will not go to the movie or Mary
will eat popcorn   
c. Mary will not eat popcorn and she
will not go to the movie

35
Sec. 3.1 Statements and Logical Connectives
Or" statements Disjunction, symbol
Let p Mary will go to the movie q Mary
will eat popcorn
ex. Write the following statements in symbolic
form   a. Mary will go to the movie or Mary will
eat popcorn
p q b. Mary will not go to the movie or
Mary will eat popcorn   
c. Mary will not eat popcorn and
she will not go to the movie

36
Sec. 3.1 Statements and Logical Connectives
Or" statements Disjunction, symbol
Let p Mary will go to the movie q Mary
will eat popcorn
ex. Write the following statements in symbolic
form   a. Mary will go to the movie or Mary will
eat popcorn
p q b. Mary will not go to the movie or
Mary will eat popcorn   
p q c. Mary will not eat
popcorn and she will not go to the movie

37
Sec. 3.1 Statements and Logical Connectives
Or" statements Disjunction, symbol
Let p Mary will go to the movie q Mary
will eat popcorn
ex. Write the following statements in symbolic
form   a. Mary will go to the movie or Mary will
eat popcorn
p q b. Mary will not go to the movie or
Mary will eat popcorn   
p q c. Mary will not eat
popcorn and she will not go to the movie
q p
38
Sec. 3.1 Statements and Logical Connectives
ex. Given p Broccoli is delicious
q Cake tastes good Write the following
symbolic statements in words. a. p
q b. p q c. (p
q) d. (p q)
39
Sec. 3.1 Statements and Logical Connectives
ex. Given p Broccoli is delicious
q Cake tastes good Write the following
symbolic statements in words. a. p
q Broccoli is delicious and cake does not taste
good. b. p q c. (p
q) d. (p q)
40
Sec. 3.1 Statements and Logical Connectives
ex. Given p Broccoli is delicious
q Cake tastes good Write the following
symbolic statements in words. a. p
q Broccoli is delicious and cake does not taste
good. b. p q Broccoli is not
delicious and cake does not taste good. c.
(p q) d. (p q)
41
Sec. 3.1 Statements and Logical Connectives
ex. Given p Broccoli is delicious
q Cake tastes good Write the following
symbolic statements in words. a. p
q Broccoli is delicious and cake does not taste
good. b. p q Broccoli is not
delicious and cake does not taste good. c.
(p q) It is not true that broccoli is
delicious or cake tastes good. d. (p
q)
42
Sec. 3.1 Statements and Logical Connectives
ex. Given p Broccoli is delicious
q Cake tastes good Write the following
symbolic statements in words. a. p
q Broccoli is delicious and cake does not taste
good. b. p q Broccoli is not
delicious and cake does not taste good. c.
(p q) It is not true that broccoli is
delicious or cake tastes good. d. (p
q) It is false that broccoli is delicious and
cake tastes good.
43
Sec. 3.1 Statements and Logical Connectives
"If - then" statements Conditional statement,
symbol If p, then q p
q
44
Sec. 3.1 Statements and Logical Connectives
"If - then" statements Conditional statement,
symbol If p, then q p
q
ex. Given p Mary is an Art student
q Liz is a News commentator
r Tim is a Doctor
45
Sec. 3.1 Statements and Logical Connectives
"If - then" statements Conditional statement,
symbol If p, then q p
q
ex. Given p Mary is an Art student
q Liz is a News commentator
r Tim is a Doctor
Write the following symbolic statements in
words a. (q p) r b. p
( r q)
46
Sec. 3.1 Statements and Logical Connectives
"If - then" statements Conditional statement,
symbol If p, then q p
q
ex. Given p Mary is an Art student
q Liz is a News commentator
r Tim is a Doctor
Write the following symbolic statements in
words a. (q p) r If Liz is not a
news commentator then Mary is an Art student or
Tim is a Doctor b. p ( r q)
47
Sec. 3.1 Statements and Logical Connectives
"If - then" statements Conditional statement,
symbol If p, then q p
q
ex. Given p Mary is an Art student
q Liz is a News commentator
r Tim is a Doctor
Write the following symbolic statements in
words a. (q p) r If Liz is not a
news commentator then Mary is an Art student or
Tim is a Doctor b. p ( r q)
48
Sec. 3.1 Statements and Logical Connectives
"If - then" statements Conditional statement,
symbol If p, then q p
q
ex. Given p Mary is an Art student
q Liz is a News commentator
r Tim is a Doctor
Write the following symbolic statements in
words a. (q p) r If Liz is not a
news commentator then Mary is an Art student or
Tim is a Doctor b. p ( r q)
49
Sec. 3.1 Statements and Logical Connectives
"If - then" statements Conditional statement,
symbol If p, then q p
q
ex. Given p Mary is an Art student
q Liz is a News commentator
r Tim is a Doctor
Write the following symbolic statements in
words a. (q p) r If Liz is not a
news commentator then Mary is an Art student, or
Tim is a Doctor b. p ( r q)
50
Sec. 3.1 Statements and Logical Connectives
"If - then" statements Conditional statement,
symbol If p, then q p
q
ex. Given p Mary is an Art student
q Liz is a News commentator
r Tim is a Doctor
Write the following symbolic statements in
words a. (q p) r If Liz is not a
news commentator then Mary is an Art student, or
Tim is a Doctor b. p ( r q) If
Mary is an Art student then Tim is a Doctor or
Liz is not a News commentator
51
Sec. 3.1 Statements and Logical Connectives
"If - then" statements Conditional statement,
symbol If p, then q p
q
ex. Given p Mary is an Art student
q Liz is a News commentator
r Tim is a Doctor
Write the following symbolic statements in
words a. (q p) r If Liz is not a
news commentator then Mary is an Art student, or
Tim is a Doctor b. p ( r q) If
Mary is an Art student then Tim is a Doctor or
Liz is not a News commentator
52
Sec. 3.1 Statements and Logical Connectives
"If - then" statements Conditional statement,
symbol If p, then q p
q
ex. Given p Mary is an Art student
q Liz is a News commentator
r Tim is a Doctor
Write the following symbolic statements in
words a. (q p) r If Liz is not a
news commentator then Mary is an Art student, or
Tim is a Doctor b. p ( r q) If
Mary is an Art student then Tim is a Doctor or
Liz is not a News commentator
53
Sec. 3.1 Statements and Logical Connectives
"If - then" statements Conditional statement,
symbol If p, then q p
q
ex. Given p Mary is an Art student
q Liz is a News commentator
r Tim is a Doctor
Write the following symbolic statements in
words a. (q p) r If Liz is not a
news commentator then Mary is an Art student, or
Tim is a Doctor b. p ( r q) If
Mary is an Art student, then Tim is a Doctor or
Liz is not a News commentator
54
Sec. 3.1 Statements and Logical Connectives
"If and Only If" statements Biconditional
statement, p iff q symbol
p ? q
55
Sec. 3.1 Statements and Logical Connectives
"If and Only If" statements Biconditional
statement, p iff q symbol
p ? q
ex. Let p Elephants can fly
q Canaries can sing

56
Sec. 3.1 Statements and Logical Connectives
"If and Only If" statements Biconditional
statement, p iff q symbol
p ? q
ex. Let p Elephants can fly
q Canaries can sing Write the following
symbolic statements in words a. p ?
q b. (q ? p)
57
Sec. 3.1 Statements and Logical Connectives
"If and Only If" statements Biconditional
statement, p iff q symbol
p ? q
ex. Let p Elephants can fly
q Canaries can sing Write the following
symbolic statements in words a. p ?
q Elephants can fly if and only if canaries
cannot sing b. (q ? p)
58
Sec. 3.1 Statements and Logical Connectives
"If and Only If" statements Biconditional
statement, p iff q symbol
p ? q
ex. Let p Elephants can fly
q Canaries can sing Write the following
symbolic statements in words a. p ?
q Elephants can fly if and only if canaries
cannot sing b. (q ? p) It is
not true that canaries can sing iff elephants
cannot fly
59
Sec. 3.1 Statements and Logical Connectives
Order of Operations Dominance of
connectives ?
60
Sec. 3.1 Statements and Logical Connectives
Order of Operations Dominance of
connectives ?
ex. p q r p
q ? r p
61
Sec. 3.1 Statements and Logical Connectives
Order of Operations Dominance of
connectives ?
ex. p q r p
q ? r p
Conditional statement
Disjunction
62
Sec. 3.1 Statements and Logical Connectives
Order of Operations Dominance of
connectives ?
ex. p q r p
q ? r p
Conditional statement
63
Sec. 3.1 Statements and Logical Connectives
Order of Operations Dominance of
connectives ?
ex. p q r p
(q r) p q ? r p
Conditional statement
64
Sec. 3.1 Statements and Logical Connectives
Order of Operations Dominance of
connectives ?
ex. p q r p
(q r) p q ? r p
Conditional statement
Biconditional statement
Conjunction
Disjunction
Negation
65
Sec. 3.1 Statements and Logical Connectives
Order of Operations Dominance of
connectives ?
ex. p q r p
(q r) p q ? r p
Conditional statement
Biconditional statement
66
Sec. 3.1 Statements and Logical Connectives
Order of Operations Dominance of
connectives ?
ex. p q r p
(q r) p q ? r p
( p q) ? (r p)
Conditional statement
Biconditional statement
67
Sec. 3.1 Statements and Logical Connectives
Quantifiers All, none, some are called
quantifiers
Be careful in finding the negation of quantifiers
!
Statement Negation All are None are Some
are Some are not
68
Sec. 3.1 Statements and Logical Connectives
Quantifiers All, none, some are called
quantifiers
Be careful in finding the negation of quantifiers
!
Statement Negation All are Some are
not None are Some are Some
are not
69
Sec. 3.1 Statements and Logical Connectives
Quantifiers All, none, some are called
quantifiers
Be careful in finding the negation of quantifiers
!
Statement Negation All are Some are
not None are Some are Some
are not All are
70
Sec. 3.1 Statements and Logical Connectives
Quantifiers All, none, some are called
quantifiers
Be careful in finding the negation of quantifiers
!
Statement Negation All are Some are
not None are Some are Some are
Some are not All are
71
Sec. 3.1 Statements and Logical Connectives
Quantifiers All, none, some are called
quantifiers
Be careful in finding the negation of quantifiers
!
Statement Negation All are Some are
not None are Some are Some are
None are Some are not All are
72
Sec. 3.1 Statements and Logical Connectives
Statement Negation All are Some
are not None are Some are Some are
None are Some are not All are
Quantifiers All, none, some are called
quantifiers
ex. Write the negation of the following
statements. a. All math books are
interesting b. No dogs have
fleas c. Some cats are
nice d. Some houses are not
homes
73
Sec. 3.1 Statements and Logical Connectives
Statement Negation All are Some
are not None are Some are Some are
None are Some are not All are
Quantifiers All, none, some are called
quantifiers
ex. Write the negation of the following
statements. a. All math books are
interesting Some math books are not
interesting b. No dogs have
fleas c. Some cats are
nice d. Some houses are not
homes
74
Sec. 3.1 Statements and Logical Connectives
Statement Negation All are Some
are not None are Some are Some are
None are Some are not All are
Quantifiers All, none, some are called
quantifiers
ex. Write the negation of the following
statements. a. All math books are
interesting Some math books are not
interesting b. No dogs have
fleas Some dogs have fleas c.
Some cats are nice d. Some
houses are not homes
75
Sec. 3.1 Statements and Logical Connectives
Statement Negation All are Some
are not None are Some are Some are
None are Some are not All are
Quantifiers All, none, some are called
quantifiers
ex. Write the negation of the following
statements. a. All math books are
interesting Some math books are not
interesting b. No dogs have
fleas Some dogs have fleas c.
Some cats are nice No cats are nice
d. Some houses are not homes
76
Sec. 3.1 Statements and Logical Connectives
Statement Negation All are Some
are not None are Some are Some are
None are Some are not All are
Quantifiers All, none, some are called
quantifiers
ex. Write the negation of the following
statements. a. All math books are
interesting Some math books are not
interesting b. No dogs have
fleas Some dogs have fleas c.
Some cats are nice No cats are nice
d. Some houses are not homes All houses
are homes
77
Sec. 3.1 Statements and Logical Connectives
Statement Negation All are Some
are not None are Some are Some are
None are Some are not All are
Quantifiers All, none, some are called
quantifiers
ex. Write the negation of the following
statements. e. No math books are
interesting f. All dogs have
fleas g. Some cats are not
nice h. Some houses are
homes
78
Sec. 3.1 Statements and Logical Connectives
Statement Negation All are Some
are not None are Some are Some are
None are Some are not All are
Quantifiers All, none, some are called
quantifiers
ex. Write the negation of the following
statements. e. No math books are
interesting Some math books are interesting
f. All dogs have fleas
g. Some cats are not nice
h. Some houses are homes
79
Sec. 3.1 Statements and Logical Connectives
Statement Negation All are Some
are not None are Some are Some are
None are Some are not All are
Quantifiers All, none, some are called
quantifiers
ex. Write the negation of the following
statements. e. No math books are
interesting Some math books are interesting
f. All dogs have fleas Some dogs
do not have fleas g. Some cats
are not nice h. Some houses
are homes
80
Sec. 3.1 Statements and Logical Connectives
Statement Negation All are Some
are not None are Some are Some are
None are Some are not All are
Quantifiers All, none, some are called
quantifiers
ex. Write the negation of the following
statements. e. No math books are
interesting Some math books are interesting
f. All dogs have fleas Some dogs
do not have fleas g. Some cats
are not nice All cats are nice
h. Some houses are homes
81
Sec. 3.1 Statements and Logical Connectives
Statement Negation All are Some
are not None are Some are Some are
None are Some are not All are
Quantifiers All, none, some are called
quantifiers
ex. Write the negation of the following
statements. e. No math books are
interesting Some math books are interesting
f. All dogs have fleas Some dogs
do not have fleas g. Some cats
are not nice All cats are nice
h. Some houses are homes No houses are homes