MATH 120
,·
CHAPTER II : Fundamentals of Logic
1 BASIC CONNECTIVES AND TRUTH TABLES
·
Definitions of Terminologies :
statement proposition declarative sentence ,
either
True or False ,
but not both
eX .
p (denotation of a statement) : Discrete Math is a required
course for sophomores
Truth Values : True or False
r = 2 + 3 = 5
Truth Values : True or False
note :
Primitive statements simple statements cannot be simplified
we usually use small alphabets to denote primitive
statements
new statements can be constructed from primitive ones in
2
ways :
1 the negation of a primitive statement
p or ~p (read as "not p" (
2 form a compound statement by logic connections
I statements formed by logic connections
Compound statements from primitive statements (
·
conjunction :
(
""
"AND" (represented by
ex . p and 9 , p q
·
Disjunction :
""
"OR" (represented by (
ex . por g or both p q
,
·
Exclusive OR :
11
represented
"
by
ex . p or
a, ph
·
Implication :
11
represented
"
by p only if q
ex .
p implies & if p then a ,
p G /
p is sufficient (condition) for 9 9 is necessary
,
(result) for P
, ·
Biconditional :
" "
represented by >
ex if and only if & iff & P
.
p , p , >&
p is sufficient and necessary for
a
* TRUTH TABLES
Truth Table for Negation
P p
T F
F T
Truth Table for Pand a ,
por a or both ,
por a
,
p -
q ps q
p E P q p q pq p q P G
F F F F F T T
F T F T T T F
T F F T T F F
T T T T F T T
we don't want true
hypothesis testing something
that is false
examples
1 .
If it is sunny today ,
then we will go to the beach
F T =
T
.
2 If it is Friday today ,
then 2 + 3 =
5
F T = T
3
. If it is Friday today ,
then 2 + 3 = 6
F F =
T
/
.
4 Let s ,
+ ,
u denote the following statements :
s : Sam goes out for a walk
t: The moon is out
U : It is snowing
, Translate the following compound statements :
a( + u) S
·
If the moon is out and it is not snowing then Sam goes
out for a walk
b (s(n + )
·
It is not the case that Sam goes out for a walk if and
if it is snowing the is
only or moon out
C + (us)
·
If the moon is out ,
then if it isn't snowing (then)
Sam goes out for a walk
REVIEW :
·
negation >
- p p
TF
F T
·
truth tables :
P q pq P E pap q P q
T T T T F T T
T F F T T F F
F T F T T T F
F F F F F T T
* From example 4C make the truth table :
&
S t U U U S t ( us)
I
T T T T T
=
F T I T T
T T F T T
F T F T F F
T F T F T T
F F T F T T
T F F T T T
F F F T F T
,·
CHAPTER II : Fundamentals of Logic
1 BASIC CONNECTIVES AND TRUTH TABLES
·
Definitions of Terminologies :
statement proposition declarative sentence ,
either
True or False ,
but not both
eX .
p (denotation of a statement) : Discrete Math is a required
course for sophomores
Truth Values : True or False
r = 2 + 3 = 5
Truth Values : True or False
note :
Primitive statements simple statements cannot be simplified
we usually use small alphabets to denote primitive
statements
new statements can be constructed from primitive ones in
2
ways :
1 the negation of a primitive statement
p or ~p (read as "not p" (
2 form a compound statement by logic connections
I statements formed by logic connections
Compound statements from primitive statements (
·
conjunction :
(
""
"AND" (represented by
ex . p and 9 , p q
·
Disjunction :
""
"OR" (represented by (
ex . por g or both p q
,
·
Exclusive OR :
11
represented
"
by
ex . p or
a, ph
·
Implication :
11
represented
"
by p only if q
ex .
p implies & if p then a ,
p G /
p is sufficient (condition) for 9 9 is necessary
,
(result) for P
, ·
Biconditional :
" "
represented by >
ex if and only if & iff & P
.
p , p , >&
p is sufficient and necessary for
a
* TRUTH TABLES
Truth Table for Negation
P p
T F
F T
Truth Table for Pand a ,
por a or both ,
por a
,
p -
q ps q
p E P q p q pq p q P G
F F F F F T T
F T F T T T F
T F F T T F F
T T T T F T T
we don't want true
hypothesis testing something
that is false
examples
1 .
If it is sunny today ,
then we will go to the beach
F T =
T
.
2 If it is Friday today ,
then 2 + 3 =
5
F T = T
3
. If it is Friday today ,
then 2 + 3 = 6
F F =
T
/
.
4 Let s ,
+ ,
u denote the following statements :
s : Sam goes out for a walk
t: The moon is out
U : It is snowing
, Translate the following compound statements :
a( + u) S
·
If the moon is out and it is not snowing then Sam goes
out for a walk
b (s(n + )
·
It is not the case that Sam goes out for a walk if and
if it is snowing the is
only or moon out
C + (us)
·
If the moon is out ,
then if it isn't snowing (then)
Sam goes out for a walk
REVIEW :
·
negation >
- p p
TF
F T
·
truth tables :
P q pq P E pap q P q
T T T T F T T
T F F T T F F
F T F T T T F
F F F F F T T
* From example 4C make the truth table :
&
S t U U U S t ( us)
I
T T T T T
=
F T I T T
T T F T T
F T F T F F
T F T F T T
F F T F T T
T F F T T T
F F F T F T
