Maharashtra Board Class 12 Maths Part 1 Chapter 1 Mathematical Logic PDF Download

Mathematical Logic

Let's Study

1. Statement and its truth value.

2. Logical connective, compound statements.

3. Truth tables negation of statements and compound statements.

4. Statement pattern, logical equivalence.

5. Tautology, contradiction and contingency.

6. Quantifiers and quantified statements, Duality.

7. Application of logic to switching circuits, switching table.

Let's Recall

1.1.1 Introduction

Mathematics is a logical subject and tries to be exact. For exactness, it requires proofs which depend upon proper reasoning. Reasoning requires logic. The word Logic is derived from the Greek word "Logos" which means reason. Therefore logic deals with the method of reasoning.

In ancient Greece the great philosopher and thinker Aristotle started study of Logic systematically. In mathematics Logic has been developed by English Philosopher and mathematician George Boole (2 November 1815 - 8 December 1864).

Language is the medium of communication of our thoughts. For communication we use sentences. In logic, we use the statements which are special sentences.

1.1.2 Statement

A statement is a declarative (assertive) sentence which is either true or false, but not both simultaneously. Statements are denoted by p, q, r, .....

Teacher's Note

A statement must have only one truth value - either true or false, never both. For example, "Two plus two equals four" is always true.

Exam Trick

Remember: Questions like "What is your name?" are NOT statements because they ask, not declare. Only sentences that declare something can be statements.

Points to Remember

A statement is a sentence that declares something.
It must be either true or false, not both at the same time.
Questions and commands are never statements.
Statements are shown using letters like p, q, r.

1.1.3 Truth Value Of A Statement

Each statement is either true or false. If a statement is true then its truth value is 'T' and if the statement is false then its truth value is F.

Following sentences are statements:

i) Sun rises in the East.

ii) \(5 \times 2 = 11\)

iii) Every triangle has three sides.

iv) Mumbai is the capital of Maharashtra.

v) Every equilateral triangle is an equiangular triangle.

vi) A natural number is an integer.

Teacher's Note

Truth value means we are checking if a statement is true or false. "The sun rises in the East" is true, so its truth value is T.

Exam Trick

Always remember: T means the statement is true, and F means the statement is false. Write only T or F as the answer, nothing else.

Points to Remember

Truth value T means the statement is true.
Truth value F means the statement is false.
Every statement has exactly one truth value.
We write T or F, not yes or no.

Following sentences are not statements:

i) Please, give your Pen.

ii) What is your name?

iii) What a beautiful place it is!

iv) How are you?

v) Do you like to play tennis?

vi) Open the window.

vii) Let us go for tea.

viii) Sit down.

Note: Interrogative, exclamatory, command, order, request, suggestion are not statements.

Teacher's Note

Commands like "Open the door" are not statements. Questions like "What time is it?" are also not statements because they don't declare anything true or false.

Exam Trick

If you see a question mark or exclamation mark or a request word like "please", it is NOT a statement in logic. Only declarative sentences with clear truth values are statements.

Points to Remember

Questions with question marks are not statements.
Commands and orders are not statements.
Requests and suggestions are not statements.
Exclamations with exclamation marks are not statements.

Consider the following:

i) \(\frac{3x}{2} - 9 = 0\)

ii) He is tall.

iii) Mathematics is an interesting subject.

iv) It is black in colour.

Let us analyse these statements:

i) For x = 6 it is true but for other than 6 it is not true.

ii) Here, we cannot determine the truth value.

For iii) and iv) the truth value varies from person to person. In all the above sentences, the truth value depends upon the situation. Such sentences are called as open sentences. Open sentence is not a statement.

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