Mathematical Logic

Mathematical Logic is Mathematical Formalization of Reasoning.

If cloth is either red or blue, and if it is not red, then logic leads us to the conclusion that it must be blue.

Sentence

A Sentence is a relatively independent Grammatical Unit

Sentences are of different types:

Type	Definition	Example
Declarative	Sentence that makes a statement, provides a fact, offers an explanation, or conveys information.	Rotational Mechanics is the easiest topic of Physics
Interrogative	A sentence that asks a question.	Is it that only smart pupil enroll in Discrete Structures Course?
Imperative	A sentence that expresses a direct command, request, invitation, warning, or instruction.	Solve the assignment and share on WhatsApp Group.
Exclamatory	Sentence that expresses strong emotion.	Oh! We have Microprocessors and Microcontrollers in next semester!
Optative	A sentence that expresses a wish, a prayer, a boon or a curse.	May this Course never End!

Statement

DECLARATIVE SENTENCE which is either TRUE or FALSE, but not both simultaneously.
True Statement are called Valid Statement.
False Statement are called Invalid Statement.

No sentence can be called a statement if it

is an exclamation
is imperative
is Interrogative
is Optative
is a request.
is a Paradox.
is Ambiguous
involves variable time such as ‘today’, ‘tomorrow’, ‘yesterday’ etc.
involves variable places such as ‘here’, ‘there’, ‘everywhere’ etc.
involves pronouns such as ‘she’, ‘he’, ‘they’

Example : [Statement highlighted in Red or Green, Sentences which are NOT Statement are italicized]

New York is in India.
Every Rhombus is Parallelogram.
Close the Door.
Turn off the Light
Moksha is the annual cultural festival of NSUT.
Today is Monday

Types of Statements

Simple: Cannot be broken down into two or more statements.
Example: 3 is an odd number.
Compound : made up of two or more simple statements.
Joined by logical connectives like and, not, or, if-then, iff etc.
Example: 371 is a composite and Armstrong number.
[The simple statements which constitutes a compound statement are called Component Statement, or sub-statements.]
Open Statement: it contain variables. When values are given, it becomes a statement.
Example:

`y\leq5`
`x` is a prime number.

Proposition and Propositional Logic

A Proposition is a SET of DECLARATIVE SENTENCES having ONE and ONLY-ONE possible values, called as TRUTH VALUES.

We have TWO Truth Values
TRUE, often labelled as 1 or T
FALSE, often labelled as 0 or F
NOTE : Open Statements are not Proposition. `y\leq5` is not a Proposition.

Example :
`P(n) : 1 + 2 + 3 + ................ + n = \frac{n(n+1)(n+2)(n+3)}{3}`

It is a proposition with Truth Value as False

`P(n) : 1 + 2 + 3 + ................ + n = \frac{n(n+1)}{2}`

It is a proposition with Truth Value as True

PROPOSITIONAL LOGIC
Simplest type of Languages to Understand Mathematical Logic.

Also called as
Zeroth Order Logic.
Statement Logic
Propositional Calculus
Sentential Logic
Sentential Calculus

(We also have First Order Logic)

REFERENCES AND FURTHER READINGS

Mathematical Logic