Summary

Full Summary of Discrete Mathematics

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Course
MATH 120 (MATH120)

Institution
Columbia College (CC )

**Discrete Mathematics Class Notes** These comprehensive class notes cover all major topics in discrete mathematics, providing a thorough foundation for understanding the principles and applications of this field. The notes include detailed explanations, examples, and proofs, making them a valua...

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Uploaded on August 1, 2024
Number of pages 113
Written in 2023/2024
Type Summary

discrete mathematics
gcd
induction
proof by induction
recursive
pascals triangle
set theory
combinatorial proof
multinomial theorem
binomial
permutation
combination
quantifiers
truth tables
contradict

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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

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