DISCRETE STRUCTURES
LOGICAL STRUCTURE
•
Logic 
study of reasoning

focuses on the relationship among
statements as opposed to the content of any
particular statement.
Logical methods are used in mathematics
to prove theorems and in computer science
to prove that programs do what they are
alleged to do.
ex.
All mathematicians wear sandals.
LOGICAL STRUCTURE
Propositions – A sentence that is either
true
or
false, but not both.
Examples:
•
1+1=3
•
2+2=4
 A factbased declaration is a proposition,
even if no one knows whether it is true.
LOGICAL STRUCTURE

A statement cannot be true or false unless
it is declarative. This excludes commands
and questions.
examples:
•
What time is it?
•
Go home.
LOGICAL STRUCTURE
Definition
Let p and q be propositions
•
The
conjunction
of p and q, denoted by
p
q
ᴧ
, is the proposition p and q
•
The
disjunction
of p and q, denoted
p
ᴠ
q
, is the proposition p or q
LOGICAL STRUCTURE
Compound proposition – combination of
propositions
Example :
p: 1+1 = 3
q: a decade is 10 years
then:
conjunction:
p
q
ᴧ
: 1+1 = 3 and a decade is
10
years.
disjunction:
p
q
ᴠ
: 1+1 = 3 or a decade is 10
LOGICAL STRUCTURE
Each sentence or the propositions has a
truth value of either true or false. The truth
values of propositions can be described by
truth tables.
LOGICAL STRUCTURE
Definition
The truth value of the compound
proposition
p
q
ᴧ
is defined by the truth table
 states that the conjunction
p
q
ᴧ
is true
provided that p and q are both true;
p
q
ᴧ
is
p
q
p
ᴧ
q
T
T
T
T
F
F
F
T
F
F
F
F
∧
LOGICAL STRUCTURE
Definition
The truth value of the compound proposition
p
q
ᴠ
is defined by the truth table
 states that the disjunction
p
q
ᴠ
is true if either p
or q or both are true; false if both p and q are false
p
q
p
q
ᴠ
T
T
T
T
F
T
F
T
T
F
F
F
LOGICAL STRUCTURE
Definition
The negation of
p
denoted by
¬p
is the proposition
not p.
The truth value of the proposition
¬p
is defined by
the truth table
In other words, the negation of a proposition has the
opposite truth value from the proposition itself.
p
¬p
T
F
F
T
LOGICAL STRUCTURE
Example
p
: Blaise Pascal invented several calculating
machines.
q
:
The first allelectronic digital computer
was constructed in the 20th century.
r
:
П was calculated to 1M decimal digits in
1954
LOGICAL STRUCTURE
•
represent the proposition:
Either Blaise Pascal invented several
calculating machines and it is not the case
that the first all electronic digital computer
was constructed in the 20th century; or П
was calculated to 1M decimal digits in 1954.
LOGICAL STRUCTURE
Definition:
If p and q are propositions, the compound
propositions
if p then q
is called conditional
proposition and is denoted by p→ q.
•
The proposition p is called the hypothesis
(or antecedent) and the proposition q is
called the conclusion (or consequent)
LOGICAL STRUCTURE
example:
p: The Math Dept. gets an additional Php
200,00.
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 Winter '99
 AverrÃ³is
 Logic, logical structure