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Jawaharlal nehru technological university anantapur college of engineering
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LECTURE NOTES
ON
MATHEMATICAL FOUNDATIONS OF
COMPUTER SCIENCE
II B. Tech I semester (JNTUH-R13)
COMPUTER SCIENCE AND ENGINEERING

UNIT-I
Mathematical Logic
Statements and notations:
A proposition or statement is a declarative sentence that is either true or false (but not both).
For instance, the following are propositions: ―Paris is in France‖ (true), ―London is in Denmark‖
(false), ―2 < 4‖ (true), ―4 = 7 (false)‖. However the following are not propositions: ―what is your
name?‖ (this is a question), ―do your homework‖ (this is a command), ―this sentence is false‖
(neither true nor false), ―x is an even number‖ (it depends on what x represents),
―Socrates‖ (it is not even a sentence). The truth or falsehood of a proposition is called its truth
value.
Connectives:
Connectives are used for making compound propositions. The main ones are the
following (p and q represent given propositions):
Name
Negation
Conjunction
Disjunction
Exclusive Or
Implication
Biconditional
Represented
¬p
Q
Q
p∧
p∨
p⊕q
p→q
p↔q
Meaning
―not p‖
―p and q‖
―p or q (or both)‖
―either p or q, but not both‖
―if p then q‖
―p if and only if q‖
Truth Tables:
Logical identity
Logical identity is an operation on one logical value, typically the value of a proposition
that produces a value of true if its operand is true and a value of false if its operand is false.
The truth table for the logical identity operator is as follows:

Logical Identity
p
p
T
T
F
F
Logical negation
Logical negation is an operation on one logical value, typically the value of a proposition
that produces a value of true if its operand is false and a value of false if its operand is true.
The truth table for NOT p (also written as ¬p or ~p) is as follows:
Logical Negation
p
¬p
T
F
F
T
Binary operations
Truth table for all binary logical operators
Here is a truth table giving definitions of all 16 of the possible truth functions of 2 binary
variables (P,Q are thus boolean variables):

P Q
0 1
2
3
4
5
6
7
8
9 10
11
12
13
14
15
T T
F F
F
F
F
F
F
F
T
T
T
T
T
T
T
T
T F
F F
F
F
T
T
T
T
F
F
F
F
T
T
T
T
F T
F F
T
T
F
F
T
T
F
F
T
T
F
F
T
T
F F
F T
F
T
F
T
F
T
F
T
F
T
F
T
F
T
where T = true and F = false.
Key:
0, false, Contradiction
1, NOR, Logical NOR
2, Converse nonimplication
3, ¬p, Negation
4, Material nonimplication
5, ¬q, Negation
6, XOR, Exclusive disjunction
7, NAND, Logical NAND
8, AND, Logical conjunction
9, XNOR, If and only if, Logical
biconditional 10, q, Projection function
11, if/then, Logical implication

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