COSC 50A

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LOGICAL SYMBOLS IN
DISCRETE
MATHEMATICS

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The rules of mathematical logic specify
methods of reasoning mathematical
statements. Greek philosopher, Aristotle,
was the pioneer of logical reasoning.
Logical reasoning provides the theoretical
base for many areas of mathematics and
consequently computer science. It has
many practical applications in computer
science like design of computing
machines, artificial intelligence, definition
of data structures for programming
languages etc.

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Propositional Logic

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Propositional Logic
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is concerned with statements to which the truth
values, “true” and “false”, can be assigned. The
purpose is to analyze these statements either
individually or in a composite manner.

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Propositional Logic
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A proposition is a collection of declarative statements that has either a truth value "true” or a truth
value "false". A propositional consists of propositional variables and connectives. We denote the
propositional variables by capital letters (A, B, etc). The connectives connect the propositional
variables.
Some examples of Propositions are given below
"Man is Mortal", it returns truth value “TRUE”
"12 + 9 = 3 – 2", it returns truth value “FALSE”
The following is not a Proposition −
"A is less than 2". It is because unless we give a specific value of A, we cannot say whether the
statement is true or false.

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Propositional Variables
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The lower case letters starting from P onwards are used to represent
propositions
Example: p: India is in Asia
q: 2 + 2 = 4

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Compound Statements
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Statements or propositional variables can be combined by means of
logical connectives (operators) to form a single statement called
compound statements.
The five logical connectives are:
Symbol Connective Name
~ Not Negation
∧ And Conjunction
∨ Or Disjunction
⟶ Implies or if...then Implication or conditional
⟷ If and only if Equivalence or biconditional

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DISJUNCTION also called OR ( )
∨
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The OR operation of two propositions A and B (written as A B) is
∨
true if at least any of the propositional variable A or B is true.
The truth table is as follows
A B A B
∨
True True True
True False True
False True True
False False False

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CONJUNCTION also called AND ( )
∧
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The AND operation of two propositions A and B (written as A B) is
∧
true if both the propositional variable A and B is true.
The truth table is as follows
A B A B
∧
True True True
True False False
False True False
False False False

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Negation (¬) (~)
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The negation of a proposition A (written as ¬A) is false when A is true
and is true when A
The truth table is as follows is false.
A ¬ A
True False
False True

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Implication / if-then ( )
→
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the proposition “if A, then B”. It is false if A is true and B is
false. The rest cases are true.
The truth table is as follows
A B A B
→
True True True
True False False
False True True
False False True

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A B is bi-conditional logical connective which is true when p and q
⇔
are same, i.e. both are false or both are true.
The truth table is as follows
If and only if ( )
⇔
A B A B
⇔
True True True
True False False
False True False
False False True

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Activity 1
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Use basic logical symbol in the problems provided below
Problem:
Let P be the statement "It is raining" and Q be the statement "The
ground is wet.“(10 points )

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Answer
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• Simple problems:
• And: Is it raining and the ground is wet? (p q)
∧
• Or: Is it raining or the ground is wet? (p q)
∨
• Negation: Is it not raining? (P ¬ q ) (~ P ~ q )
• Implication: If it is raining, then the ground is wet. (p q)
⟹
• If and only if: It is raining if and only if the ground is wet. (p q)
⇔
• If and only if: If It is raining and only if the ground is wet (p q)
⇔