Download Chapter 1-Part III

CHAPTER 1
SETS, FUNCTIONS,
ELEMENTARY
LOGIC & BOOLEAN
ALGEBRAS
BY: MISS FARAH ADIBAH ADNAN
IMK
CHAPTER OUTLINE: PART III
1.3 ELEMENTARY LOGIC
1.3.1 INTRODUCTION
1.3.2 PROPOSITION
1.3.3 COMPOUND STATEMENTS
1.3.4 LOGICAL CONNECTIVES
1.3.5 CONDITIONAL STATEMENT
1.3.6 PROPOSITIONAL EQUIVALENCES
1.3 ELEMENTARY LOGIC
1.3.1 INTRODUCTION
Logic – used to distinguish between valid and
invalid mathematical arguments.
 Application in computer science – design
computer circuits, construction of computer
program, verification of the correctness of
programs.
 Basic building blocks - Prepositions

1.3.2 PROPOSITION
Proposition – is a declarative sentence either
true or false, but not both.
 Eg:
1)
Washington, D.C., is the capital of the
United States of America.
2)
1+1=2
3)
What time is it?
4)
Read this carefully.
5)
x+1=2

Letters are used to denote prepositions – p, q, r,
s.

1.3.3 COMPOUND STATEMENTS


Many mathematical statements are constructed by
combining one or more propositions.
Eg:
John is smart or he studies every night.
Fundamental property of a compound proposition:
The truth value is determined by the truth value of
its subpropositions, together with the way they are
connected to form compound proposition.
1.3.4 LOGICAL CONNECTIVES
1) Not (negation) : ~ / 
Let p be a proposition. The negation of p is denoted by
 p, and read as “not p”.
-Eg:
Find the negation of the preposition “Today is
Friday”.
The Truth Table for the Negation of a Preposition
p
p
T
F
F
T
1.3.4 LOGICAL CONNECTIVES
2) And (conjunction) : 
Let p and q be prepositions. The preposition of “p and
q” - denoted p  q , is TRUE when BOTH p and q are
true and otherwise is FALSE.
The Truth Table for the Conjunction of Two
Prepositions
pq
p
q
T
T
T
T
F
F
F
T
F
F
F
F
1.3.4 LOGICAL CONNECTIVES
3) Or (disjunction) : 
Let p and q be prepositions. The preposition of “p or q”
- denoted p  q , is FALSE when BOTH p and q are
FALSE and TRUE otherwise.
The Truth Table for the Disjunction of Two
Prepositions
p
q
pq
T
T
T
T
F
T
F
T
T
F
F
F
EXAMPLE 1.1
Consider the following statements, and determine
whether it is true or false.
1) Ice floats in water and 2 + 2 = 4
2) China is in Europe and 2 + 2 = 4
3) 5 – 3 = 1 or 2 x 2 = 4
EXAMPLE 1.2
Let p and q be the following propositions:
p = It is below freezing
q = It is snowing
Translate the following into logical notation, using
p and q and logical connectives.
(a)
(b)
(c)
(d)
It is below freezing and snowing
It is below freezing but not snowing
It is not below freezing and it is not snowing
It is either snowing or below freezing (or both)
1.3.5 CONDITIONAL STATEMENTS
1) Conditional Statement/ Implication
Let p and q be a preposition. The implication p  q is the
preposition that is FALSE when p is true, q is false.
Otherwise is TRUE.
p = hypothesis/antecedent/premise
q = conclusion/consequence
Express: “ if p, then q”, “q when p”, “p implies q”
The Truth Table for the Implication (p  q)
p
q
pq
T
T
T
T
F
F
F
T
T
F
F
T
1.3.5 CONDITIONAL STATEMENTS
2) Equivalence/ Biconditional
Let p and q be a preposition. The biconditional p  q is the
preposition that is TRUE when p and q have the same truth
values, and FALSE otherwise.
Express: “ p if and only if q”
The Truth Table for the Biconditional (p  q )
p
q
pq
T
T
T
T
F
F
F
T
F
F
F
T
CONVERSE, CONTRAPOSITIVE
CONVERSE : the implication of q  p is called
converse of p  q
CONTRAPOSITIVE : the contrapositive of p  q is
the implication  q   p
Example: refer textbook
1.3.6 PROPOSITIONAL EQUIVALENCES
Tautology
 A compound proposition that is always TRUE, no
matter what the truth values of the propositions that
occur in it.
 Contains only “T” in the last column of their truth
table.
Contradiction
 A compound proposition that is always FALSE.
 Contains only “F” in the last column of their truth
table.
1.3.6 PROPOSITIONAL EQUIVALENCES
Example:
p
p
p  p
p  p
T
F
T
F
F
T
T
F
1.3.6 PROPOSITIONAL EQUIVALENCES
Contingency
 A proposition that is neither a tautology nor a
contradiction
Example: refer text book
1.3.6 PROPOSITIONAL EQUIVALENCES
Logically Equivalent
Two propositions p and q are said to be logically
equivalent, or simply equivalent or equal, denoted by
pq
if they have identical truth tables.
Example: Find the truth tables of  (p  q) and p  q
p
q
p^q
-(p^q)
p
q
-p
-q
-p v -q
T
T
T
F
T
T
F
F
F
T
F
F
T
T
F
F
T
T
F
T
F
T
F
T
T
F
T
F
F
F
T
F
F
T
T
T