Discrete mathematics Chapter1 presentation.ppt

Discrete Mathematics
Chapter 1:
Fundamentals of Logic
1
Presented By,
Nandini S R
Department of Computer Science

1.1 Propositions
• A proposition is a declarative sentence (a sentence that declares a
fact) that is either true or false, but not both.
• Are the following sentences propositions?
• Delhi is the capital of India.
• Read this carefully.
• 1+2=3
• x+1=2
• What time is it?
• 3 is a prime number.
• Bengaluru is in Karnataka.
2

1.1 Propositions
• Propositional Variables – variables that represent propositions: p, q,
r, s
• E.g. Proposition p – “3 is a prime number.”
• Truth values – The truth or falsity of a proposition
If proposition is true, its truth value is 1 (T)
If proposition is false, its truth value is 0 (F)
3

Logical connectives
• Propositions used with words as not , and, if….then and if and only if,
gives new propositions called compound propositions.
• Words such as not , and, if….then and if and only if are called logical
connectives.
• Propositions which do not contain any logical connective are called
Simple propositions.
4

Negation
• Examples
• Find the negation of the proposition “Today is Friday.”
• Find the negation of the proposition “3 is a prime number.”
5
Let p be a proposition. The negation of p, denoted by ¬p.
The proposition ¬p is read “not p.” The truth value of the negation of p, ¬p
is the opposite of the truth value of p.
Solution: The negation is “Today is not Friday.” or “It is not
Friday today.”
Solution: The negation is “3 is not a prime number.”

Negation
The Truth Table for the
Negation of a Proposition.
p ¬p
T
F
F
T
6

Conjuction
• Examples
• Find the conjunction of the propositions p and q where p is the
proposition “9 is a prime number.” and q is the proposition “2+5=7”, and
the truth value of the conjunction.
7
Let p and q be propositions. The conjunction of p and q, denoted by
p Λ q, is the proposition “p and q”. The conjunction p Λ q is true when
both p and q are true and is false otherwise.
Solution: The conjunction is the proposition “9 is a prime number
and 2+5=7.” The proposition is False.

1.1 Propositional Logic
• E.g Find the conjunction of the propositions p and q where p is the proposition “4
is a prime number.” or q is the proposition “2+4=7”, and the truth value of the
conjunction
• Solution: The conjunction is the proposition “4 is a prime number and
2+4=7.” The proposition is False.
8
DEFINITION 3
Let p and q be propositions. The disjunction of p and q, denoted by p ν
q, is the proposition “p or q”. The conjunction p ν q is false when both
p and q are false and is true otherwise.

Exclusive Or (XOR) of
Two Propositions.
p q p q
T T
T F
F T
F F
F
T
T
F

The Truth Table for
the Conjunction of
Two Propositions.
p q p Λ q
T T
T F
F T
F F
T
F
F
F
The Truth Table for
the Disjunction of
Two Propositions.
p q p ν q
T T
T F
F T
F F
T
T
T
F
9
DEFINITION 4
Let p and q be propositions. The exclusive or of p and q, denoted by p q,
is the proposition that is true when exactly one of p and q is true and is
false otherwise.




10
DEFINITION 5
Let p and q be propositions. The conditional statement p → q, is the
proposition “if p, then q.” The conditional statement is false when p is
true and q is false, and true otherwise. In the conditional statement p
→ q, p is called the hypothesis (or antecedent or premise) and q is
called the conclusion (or consequence).
Conditional Statements
 A conditional statement is also called an implication.
 Example: “If I am elected, then I will lower taxes.” p → q
implication:
elected, lower taxes. T T | T
not elected, lower taxes. F T | T
not elected, not lower taxes. F F | T
elected, not lower taxes. T F | F

• Example:
• Let p be the statement “Maria learns discrete mathematics.” and q the
statement “Maria will find a good job.” Express the statement p → q as a
statement in English.
11
Solution: Any of the following -
“If Maria learns discrete mathematics, then she will find a
good job.

• Other conditional statements:
• Converse of p → q : q → p
• Contrapositive of p → q : ¬ q → ¬ p
• Inverse of p → q : ¬ p → ¬ q
12

• p ↔ q has the same truth value as (p → q) Λ (q → p)
• “if and only if” can be expressed by “iff”
• Example:
• Let p be the statement “You can take the flight” and let q be the
statement “You buy a ticket.” Then p ↔ q is the statement
“You can take the flight if and only if you buy a ticket.”
Implication:
If you buy a ticket you can take the flight.
If you don’t buy a ticket you cannot take the flight.
13
DEFINITION 6
Let p and q be propositions. The biconditional statement p ↔ q is the
proposition “p if and only if q.” The biconditional statement p ↔ q is
true when p and q have the same truth values, and is false otherwise.
Biconditional statements are also called bi-implications.

Biconditional p ↔ q.
p q p ↔ q
T T
T F
F T
F F
T
F
F
T
14

• We can use connectives to build up complicated compound propositions
involving any number of propositional variables, then use truth tables to
determine the truth value of these compound propositions.
• Example: Construct the truth table of the compound proposition
(p ν ¬q) → (p Λ q).
The Truth Table of (p ν ¬q) → (p Λ q).
p q ¬q p ν ¬q p Λ q (p ν ¬q) → (p Λ q)
T T
T F
F T
F F
F
T
F
T
T
T
F
T
T
F
F
F
T
F
T
F
15
Truth Tables of Compound Propositions

• We can use parentheses to specify the order in which logical operators in a
compound proposition are to be applied.
• To reduce the number of parentheses, the precedence order is defined for
logical operators.
Precedence of Logical Operators.
Operator Precedence
¬ 1
Λ
ν
2
3
→
↔
4
5
16
Precedence of Logical Operators
E.g. ¬p Λ q = (¬p ) Λ q
p Λ q ν r = (p Λ q ) ν r
p ν q Λ r = p ν (q Λ r)

• Example: How can this English sentence be translated into a logical
expression?
“You can access the Internet from campus only if you are a
computer science major or you are not a freshman.”
17
Solution: Let p, q, and r represent “You can access the Internet from
campus,” “You are a computer science major,” and “You are
a freshman.” The sentence can be translated into:
p → (q ν ¬r).

• Computers represent information using bits.
• A bit is a symbol with two possible values, 0 and 1.
• By convention, 1 represents T (true) and 0 represents F (false).
• A variable is called a Boolean variable if its value is either true or false.
• Bit operation – replace true by 1 and false by 0 in logical operations.
Table for the Bit Operators OR, AND, and XOR.
x y x ν y x Λ y x y
0
0
1
1
0
1
0
1
0
1
1
1
0
0
0
1
0
1
1
0

18
Logic and Bit Operations

• Example: Find the bitwise OR, bitwise AND, and bitwise XOR of the bit string
01 1011 0110 and 11 0001 1101.
19
DEFINITION 7
A bit string is a sequence of zero or more bits. The length of this string
is the number of bits in the string.
Solution:
01 1011 0110
11 0001 1101
-------------------
11 1011 1111 bitwise OR
01 0001 0100 bitwise AND
10 1010 1011 bitwise XOR

1.2 Propositional Equivalences
Examples of a Tautology and a Contradiction.
p ¬p p ν ¬p p Λ ¬p
T
F
F
T
T
T
F
F
20
DEFINITION 1
A compound proposition that is always true, no matter what the truth
values of the propositions that occurs in it, is called a tautology. A
compound proposition that is always false is called a contradiction. A
compound proposition that is neither a tautology or a contradiction is
called a contingency.
Introduction

• Compound propositions that have the same truth values in all possible cases
are called logically equivalent.
• Example: Show that ¬p ν q and p → q are logically equivalent.
Truth Tables for ¬p ν q and p → q .
p q ¬p ¬p ν q p → q
T
T
F
F
T
F
T
F
F
F
T
T
T
F
T
T
T
F
T
T 21
DEFINITION 2
The compound propositions p and q are called logically equivalent if p ↔
q is a tautology. The notation p ≡ q denotes that p and q are logically
equivalent.
Logical Equivalences

• In general, 2n rows are required if a compound proposition involves n
propositional variables in order to get the combination of all truth values.
• See page 24, 25 for more logical equivalences.
22

• Example: Show that ¬(p → q ) and p Λ ¬q are logically equivalent.
Solution:
¬(p → q ) ≡ ¬(¬p ν q) by example on slide 21
≡ ¬(¬p) Λ ¬q by the second De Morgan law
≡ p Λ ¬q by the double negation law
• Example: Show that (p Λ q) → (p ν q) is a tautology.
Solution: To show that this statement is a tautology, we will use logical
equivalences to demonstrate that it is logically equivalent to T.
(p Λ q) → (p ν q) ≡ ¬(p Λ q) ν (p ν q) by example on slide 21
≡ (¬ p ν ¬q) ν (p ν q) by the first De Morgan law
≡ (¬ p ν p) ν (¬ q ν q) by the associative and
communicative law for disjunction
≡ T ν T
≡ T
• Note: The above examples can also be done using truth tables.
23
Constructing New Logical Equivalences

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