1. LECTURE 1

2. Disrete
mathematics
and its
application by
rosen
7th edition
THE FOUNDATIONS:
LOGIC AND PROOFS
1.1 PROPOSITIONAL
LOGIC

3.  A proposition is a declarative sentence (that is, a sentence
that declares a fact) that is either true or false, but not both
 1 + 1 = 2 (true)
 4 + 9 = 13 (true)
 Islamabad is capital of Pakistan (true)
 Karachi is the largest city of Pakistan (true)
 100+9 = 111 (false)
 Some sentences are not prepositions
 Where is my class? (un decelerated sentence)
 What is the time by your watch? (un decelerated sentence)
 x + y = ? ( will be prepositions when value is assigned)
 Z +w * r = p
PROPOSITIONS

4.  We use letters to denote propositional variables (or statement
variables).
 The truth value of a proposition is true, denoted by T, if it is a
true proposition.
 The truth value of a proposition is false, denoted by F, if it is a
false proposition.
 Many mathematical statements are constructed by combining
one or more propositions. They are called compound
propositions, are formed from existing propositions using
logical operators.
PROPOSITIONS

5.  Definition: Let p be a proposition. The negation of p, denoted
by￢p (also denoted by p), is the statement “It is not the case
that 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.
 Also denoted as “ ′ ”
 Examples:
 p := Sir PC is running Windows OS
 ￢p := sir PC is not running Windows OS
 p := a + b = c
 p := a + b ≠ c
NEGATION
The Truth Table for the Negation of a Proposition
p ￢p
T
F
F
T

6.  Definition: Let p and q be propositions. The conjunction of p
and q, denoted by p ∧ q, is the proposition “p and q.” The
conjunction p ∧ q is true when both p and q are true and is
false otherwise.
 Also known as UNION, AND, BIT WISE AND, AGREGATION
 Denoted as ^ , &, AND
CONJUNCTION
The Truth Table for the conjunction of a Proposition
p q p ^ q
T T
T F
F T
F F
T
F
F
F

7.  Definition: Let p and q be propositions. The disjunction of p
and q, denoted by p ∨ q, is the proposition “p or q.” The
disjunction p ∨ q is false when both p and q are false and is
true otherwise.
 Also known as OR, BIT WISE OR, SEGREGATION
 Denoted as v , || , OR
DISJUNCTION
The Truth Table for the conjunction of a Proposition
p q p v q
T T
T F
F T
F F
T
T
T
F

8.  Definition: 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.
 Also known as ZORING
 Denoted as XOR , Ex OR, ⊕
EXCLUSIVE OR
The Truth Table for the conjunction of a Proposition
p q p ⊕ q
T T
T F
F T
F F
F
T
T
F

9.  Let p and q be propositions. The conditional statement p → q
is the proposition “if p, then q.” The conditional statement p
→ q 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).
 Denoted by 
CONDITIONAL STATEMENT
The Truth Table for the conjunction of a Proposition
p q p → q
T T
T F
F T
F F
T
F
T
T

10.  The proposition q → p is called the converse of p → q.
 The converse, q → p, has no same truth value as p → q for all
cases.
 Formed from conditional statement.
CONVERSE

11.  The contrapositive of p → q is the proposition ￢q →￢p.
 only the contrapositive always has the same truth value as p
→ q.
 The contrapositive is false only when ￢p is false and ￢q is
true.
 Formed from conditional statement.
CONTRAPOSITIVE
The Truth Table for the CONTRAPOSITIVE of a Proposition
p q ￢p ￢q ￢p → ￢q
T T F F
T F F T
F T T F
F F T T
T
T
F
T

12.  Formed from conditional statement.
 The proposition ￢p →￢q is called the inverse of p → q.
 The converse, q → p, has no same truth value as p → q for all
cases.
INVERSE

13.  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
The Truth Table for the CONTRAPOSITIVE of a Proposition
q p q ↔ p
T T
T F
F T
F F
T
F
F
T

14.  Definition: When more that one above defined preposition
logic combines it is called as compound preposition.
 Example:
 (p^q)v(p’)
 (p ⊕ q) ^ (r v s)
COMPOUND PROPOSITIONS

15.  (p ∨￢q) → (p ∧ q)
COMPOUND PROPOSITIONS (TRUTH
TABLE)
The Truth Table of (p ∨￢q) → (p ∧ q)
p q ￢q p ∨￢q p ∧ q (p ∨￢q) → (p ∧ q)
T
T
F
F
T
F
T
F
F
T
F
T
T
T
F
T
T
F
F
F
T
F
T
F

16. Precedence of Logical Operators.
Operator Precedence
￢ 1
^ 2
v 3
→ 4
↔ 5
XOR 6
PRECEDENCE OF LOGICAL OPERATORS

17.  Computers represent information using bits
 A bit is a symbol with two possible values, namely, 0 (zero)
and 1 (one).
 A bit can be used to represent a truth value, because there
are two truth values, namely, true and false.
 1 bit to represent true and a 0 bit to represent false. That is,
1 represents T (true), 0 represents F (false).
 A variable is called a Boolean variable if its value is either
true or false. Consequently, a Boolean variable can be
represented using a bit.
LOGIC AND BIT OPERATIONS
Truth Value Bit
T 1
F 0

18.  Computer bit operations correspond to the logical
connectives.
 By replacing true by a one and false by a zero in the truth
tables for the operators ∧ (AND) , ∨ (OR) , and ⊕ (XOR) , the
tables shown for the corresponding bit operations are
obtained.
LOGIC AND BIT OPERATIONS
Table for the Bit Operators OR, AND, and XOR.
p q p ^ q p v q p XOR q
0
0
1
1
0
1
0
1
0
0
0
1
0
1
1
1
0
1
1
0

19.  01 1011 0110
11 0001 1101
11 1011 1111 bitwise OR
01 0001 0100 bitwise AND
10 1010 1011 bitwise XOR
 11 1010 1110
11 0001 1101
11 1011 1111 bitwise OR
11 0000 1100 bitwise AND
00 1010 0011 bitwise XOR
BITWISE OR, BITWISE AND, AND BITWISE
XOR