Maharashtra Board Class 12 Maths Commerce Part I Chapter 1 Mathematical logic PDF Download

Mathematical Logic

Let's Study

Statement

Logical connectives

Quantifiers and quantified statements

Statement patterns and logical equivalence

Algebra of statements

Venn diagrams

Introduction

Mathematics is an exact science. Every statement must be precise. There has to be proper reasoning in every mathematical proof. Proper reasoning involves Logic. Logic related to mathematics has been developed over last 100 years or so. The axiomatic approach to logic was first propounded by the English philosopher and mathematician George Boole. Hence it is known as Boolean logic or mathematical logic or symbolic logic.

The word 'logic' is derived from the Greek word 'Logos' which means reason. Thus Logic deals with the method of reasoning. Aristotle (382-322 B.C.), the great philosopher and thinker laid down the foundations of study of logic in a systematic form. The study of logic helps in increasing one's ability of systematic and logical reasoning and develop the skill of understanding validity of statements.

1.1 Statement

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

For example:

i) 2 is a prime number.

ii) Every rectangle is a square.

iii) The Sun rises in the West.

iv) Mumbai is the capital of Maharashtra.

Truth value of a statement

A statement is either true or false. The truth value of a 'true' statement is denoted by T (TRUE) and that of a false statement is denoted by F (FALSE).

Example 1: Observe the following sentences.

i) The Sun rises in the East.

ii) The square of a real number is negative.

iii) Sum of two odd numbers is odd.

iv) Sum of opposite angles in a cyclic rectangle is 180°.

Here, the truth value of statements (i) and (iv) is T, and that of (ii) and (iii) is F.

Note: The sentences like exclamatory, interrogative, imperative are not considered as statements.

Example 2: Observe the following sentences.

i) May God bless you!

ii) Why are you so unhappy?

iii) Remember me when we are parted.

iv) Don't ever touch my phone.

v) I hate you!

vi) Where do you want to go today?

The above sentences cannot be assigned truth values, so none of them is a statement.

The sentences (i) and (v) are exclamatory.

The sentences (ii) and (vi) are interrogative.

The sentences (iii) and (iv) are imperative.

Teacher's Note

A statement must be either true or false, never both. Think of Aadhaar number - it is either valid or invalid for a person, not both at the same time.

Exam Trick

Remember: Exclamatory, interrogative, and imperative sentences are NOT statements. Only declarative sentences with clear true or false values are statements.

Points to Remember

A statement is a sentence that is definitely true or definitely false.
Exclamatory sentences like "What a beautiful day!" are not statements.
Interrogative sentences like "Why are you here?" are not statements.
Imperative sentences like "Close the door" are not statements.
Truth value is T for true statements and F for false statements.

Open sentences

An open sentence is a sentence whose truth can vary according to some conditions which are not stated in the sentence.

Example 3: Observe the following.

i) x + 4 = 8

ii) Chinese food is very tasty

Each of the above sentences is an open sentence, because truth of (i) depends on the value of x; if x = 4, it is true and if x ≠ 4, it is false and that of (ii) varies as degree of tasty food varies from individual to individual.

Note:

i) An open sentence is not considered a statement in logic.

ii) Mathematical identities are true statements.

For example:

a + 0 = 0 + a = a, for any real number a.

Activity

Determine whether the following sentences are statements in logic and write down the truth values of the statements.

(Note: Complete the above table)

EXERCISE 1.1

State which of the following sentences are statements. Justify your answer if it is a statement. Write down its truth value.

i) A triangle has 'n' sides

ii) The sum of interior angles of a triangle is 180°

iii) You are amazing!

iv) Please grant me a loan.

v) 4 is an irrational number.

vi) x² − 6x + 8 = 0 implies x = 4 or x = 2.

Teacher's Note

Open sentences have variables and their truth depends on those variables. For example, "x + 2 = 5" is true only when x = 3, so it is not a statement.

Exam Trick

Remember: If a sentence has a variable (like x, y, n) whose value is not fixed, it is an open sentence, NOT a statement. Ask yourself: "Can I say this is definitely true or definitely false?"

Points to Remember

Open sentences contain variables whose values are not given.
Open sentences are NOT statements because truth value depends on the variable values.
Mathematical identities like (a + b)² = a² + 2ab + b² are statements because they are always true.
Every statement is either true or false, never both, never neither.

This is a preview of the first 3 pages. To get the complete book, click below.