UNIT 4 – MATHEMATICAL LOGIC

1. Statement

Definition

A statement is a sentence that has a definite truth value, meaning it must be either True or False, but not both.

Explanation

In mathematical logic, a statement is used to express facts. If we can clearly decide whether a sentence is correct or incorrect, then it is called a statement.

Questions, commands, or unclear sentences are not statements because they do not have fixed truth values.

Examples

1. India is in Asia. → True
2. 5 + 2 = 10. → False
3. The Earth revolves around the Sun. → True

Not Statements

• What is your name? (Question)
• Shut the door. (Command)

2. Proposition

Definition

A proposition is a declarative statement that is either true or false.

Explanation

In logic, proposition and statement are almost the same. We usually represent propositions by letters like P, Q, R.

Example

P = Delhi is the capital of India. → True Q = 8 is smaller than 2. → False

So both P and Q are propositions.

3. Negation

Definition

Negation means the opposite of a statement.

Symbol = ¬P

Explanation

If a statement is true, its negation becomes false. If a statement is false, its negation becomes true.

Negation simply reverses the truth value.

Example

P = It is raining. ¬P = It is not raining.

If P is true → ¬P becomes false. If P is false → ¬P becomes true.

4. Conjunction

Definition

Conjunction is formed when two statements are joined by AND.

Symbol = P ∧ Q

Explanation

Conjunction becomes true only when both statements are true. If even one statement is false, the whole conjunction becomes false.

Example

P = I study. Q = I pass.

P ∧ Q = I study AND I pass.

Truth Table

P | Q | P∧Q T | T | T T | F | F F | T | F F | F | F

So conjunction is true only when both are true.

5. Disjunction

Definition

Disjunction is formed when two statements are joined by OR.

Symbol = P ∨ Q

Explanation

Disjunction becomes true if at least one statement is true. It becomes false only when both statements are false.

Example

P = I drink tea. Q = I drink coffee.

P ∨ Q = I drink tea OR coffee.

Truth Table

P | Q | P∨Q T | T | T T | F | T F | T | T F | F | F

6. Implication

Definition

Implication is a logical statement of the form If P, then Q.

Symbol = P → Q

Explanation

It means when P happens, Q should follow.

This is widely used in mathematics and programming logic.

Example

P = I study. Q = I pass.

P → Q = If I study, then I pass.

Truth Table

P | Q | P→Q T | T | T T | F | F F | T | T F | F | T

Important: Only T → F is false.

7. Bi-conditional

Definition

Bi-conditional means if and only if.

Symbol = P ↔ Q

Explanation

It becomes true when both statements have the same truth value.

Example

P = You are eligible. Q = You are 18+.

P ↔ Q = You are eligible if and only if you are 18+.

Truth Table

P | Q | P↔Q T | T | T T | F | F F | T | F F | F | T

8. Truth Table

Definition

A truth table is a table showing all possible true and false combinations of logical statements.

Explanation

Truth tables help us check whether an expression is always true, false, or conditionally true.

Example

P | Q | P∧Q | P∨Q | P→Q T | T | T | T | T T | F | F | T | F F | T | F | T | T F | F | F | F | T

9. Converse

Definition

Converse means interchanging P and Q.

If original statement is: P → Q

Then converse is: Q → P

Example

Original: If I study, I pass. Converse: If I pass, I study.

10. Inverse

Definition

Inverse means negating both P and Q.

If original statement is: P → Q

Then inverse is: ¬P → ¬Q

Example

Original: If I study, I pass. Inverse: If I do not study, I do not pass.

11. Contrapositive

Definition

Contrapositive means interchange and negate both statements.

If original statement is: P → Q

Then contrapositive is: ¬Q → ¬P

Example

Original: If I study, I pass. Contrapositive: If I do not pass, I did not study.

12. Tautology

Definition

A tautology is a statement that is always true, no matter what the truth values are.

Explanation

Its truth value is always true in every possible case.

Example

P ∨ ¬P

If P = True → result True If P = False → ¬P becomes True

So always true.

13. Contradiction

Definition

A contradiction is a statement that is always false.

Explanation

No matter what values are used, result remains false.

Example

P ∧ ¬P

A statement cannot be both true and false together.

So always false.

14. Logical Equivalence

Definition

Two logical statements are logically equivalent if they always produce the same truth value.

Explanation

Even if expressions look different, if truth tables are same, they are equivalent.

Example

P → Q is logically equivalent to ¬P ∨ Q

Both give same result.

15. De Morgan’s Law

Definition

De Morgan’s laws are used to simplify logical expressions.

They change AND into OR, and OR into AND, while negating terms.

First Law

NOT (P AND Q) = (NOT P) OR (NOT Q)

Example

P = I study Q = I pass

NOT (I study AND I pass) = I do not study OR I do not pass

Second Law

NOT (P OR Q) = (NOT P) AND (NOT Q)

Example

P = I play cricket Q = I play football

NOT (P OR Q) = I do not play cricket AND I do not play football

Quick Revision

• Statement = sentence with true/false value
• Proposition = declarative statement
• Negation = opposite
• Conjunction = AND
• Disjunction = OR
• Implication = If-Then
• Bi-conditional = If and only if
• Converse = interchange
• Inverse = negate
• Contrapositive = interchange + negate
• Tautology = always true
• Contradiction = always false
• De Morgan’s law = simplify logic expressions