Topics
Mathematical Logic
- Statements and Truth Values in Mathematical Logic
- Logical Connectives
- Tautology, Contradiction, and Contingency
- Quantifier, Quantified and Duality Statements in Logic
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits
- Overview of Mathematical Logic
Matrices
Trigonometric Functions
Pair of Straight Lines
Vectors
Line and Plane
Linear Programming
Differentiation
- Introduction & Derivatives of Some Standard Functions
- Derivative of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Function
- Logarithmic Differentiation
- Derivatives of Implicit Functions
- Derivatives of Parametric Functions
- Higher Order Derivatives
- Overview of Differentiation
Applications of Derivatives
- Applications of Derivatives in Geometry
- Derivatives as a Rate Measure
- Approximations
- Rolle's Theorem
- Lagrange's Mean Value Theorem (LMVT)
- Increasing and Decreasing Functions
- Maxima and Minima
- Overview of Applications of Derivatives
Indefinite Integration
Definite Integration
- Definite Integral as Limit of Sum
- Integral Calculus
- Methods of Evaluation and Properties of Definite Integral
- Overview of Definite Integration
Application of Definite Integration
- Application of Definite Integration
- Area Bounded by Two Curves
- Overview of Application of Definite Integration
Differential Equations
Probability Distributions
- Random Variables
- Probability Distribution of Discrete Random Variables
- Probability Distribution of a Continuous Random Variable
- Variance of a Random Variable
- Expected Value and Variance of a Random Variable
- Overview of Probability Distributions
Binomial Distribution
Definition: 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, ....
Open Statement:
An open sentence is a sentence whose truth can vary depending on conditions not stated in the sentence.
Definition: 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.
Definition: Logical Connectives
The words or phrases which are used to connect two statements are called logical connectives.
Eg. 'and', 'or', 'if ..... then', 'if and only if ', 'not''.
Definition: Simple and Compound Statements
A statement which cannot be split further into two or more statements is called a simple statement. If a statement is the combination of two or more simple statements, then it is called a compound statement.
Key Points: Logical Connectives
| Connective | Symbol | Name | True When |
|---|---|---|---|
| and | ∧ | Conjunction | Both true |
| or | ∨ | Disjunction | At least one true |
| if...then | → | Conditional | False only when T → F |
| iff | ↔ | Biconditional | Same truth values |
| not | ~ | Negation | Opposite value |
Note: ~ (~ p) = p
Definition: Logical Equivalence
Two statement patterns are said to be equivalent if their truth tables are identical. If statement patterns A and B are equivalent, we write it as A ≡ B.
Key Points: Tautology, Contradiction and Contingency
| Type | Meaning |
|---|---|
| Tautology | Always True |
| Contradiction | Always False |
| Contingency | Sometimes T, Sometimes F |
Key Points: Quantifiers and Quantified Statements
| Symbol | Meaning |
|---|---|
| ∀ | For all (Universal quantifier) |
| ∃ | There exists (Existential quantifier) |
Quantified statement: An open sentence with a quantifier becomes a statement and is called a quantified statement.
Key Points: Negation of Compound Statements
| Type | Given Statement | Negation | Symbolic Form |
|---|---|---|---|
| Negation of Conjunction | p ∧ q | Not p or Not q | ~(p ∧ q) ≡ ~p ∨ ~q |
| Negation of Disjunction | p ∨ q | Not p and Not q | ~(p ∨ q) ≡ ~p ∧ ~q |
| Negation of Implication | p → q | p and Not q | ~(p → q) ≡ p ∧ ~q |
| Negation of Biconditional | p ↔ q | (p and Not q) or (q and Not p) | ~(p ↔ q) ≡ (p ∧ ~q) ∨ (q ∧ ~p) |
| Negation of Quantified Statement | ∀ x P(x) / ∃ x P(x) | Replace “all” by “some” and vice versa, and negate P(x) | ~(∀ x P(x)) ≡ ∃x ~P(x) ~(∃x P(x)) ≡ ∀x ~P(x) |
Key Points: Algebra of Statements
| Sr. No. | Law Name | Logical Form |
|---|---|---|
| 1 | Idempotent Law | p ∧ p ≡ p p ∨ p ≡ p |
| 2 | Commutative Law | p ∧ q ≡ q ∧ p p ∨ q ≡ q ∨ p |
| 3 | Associative Law | p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r ≡ p ∧ q ∧ r p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r ≡ p ∨ q ∨ r |
| 4 | Distributive Law | p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) |
| 5 | De Morgan’s Laws | ~(p ∧ q) ≡ ~p ∨ ~q ~(p ∨ q) ≡ ~p ∧ ~q |
| 6 | Identity Laws | p ∧ T ≡ p p ∨ F ≡ p p ∧ F ≡ F p ∨ T ≡ T |
| 7 | Complement Laws | p ∧ ~p ≡ F p ∨ ~p ≡ T |
| 8 | Absorption Laws | p ∨ (p ∧ q) ≡ p p ∧ (p ∨ q) ≡ p |
| 9 |
Conditional Law |
p → q ≡ ~p ∨ q |
| 10 | Biconditional Law |
p ↔ q ≡ (p → q) ∧ (q → p) ≡ (~p ∨ q) ∧ (~q ∨ p) |
Key Points: Switching Circuits
| Circuit Type | Logical Form |
|---|---|
| Series | p ∧ q |
| Parallel | p ∨ q |
Switch ON = 1
Switch OFF = 0
Key Points: Converse, Inverse, Contrapositive
For p → q:
| Type | Form |
|---|---|
| Converse | q → p |
| Inverse | ∼p → ∼q |
| Contrapositive | ∼q → ∼p |
Definition: Duality
Two compound statements s1 and s2 are said to be duals of each other if one can be obtained from the other by:
-
Replacing ∧ (and) by ∨ (or)
-
Replacing ∨ (or) by ∧ (and)
-
Replacing T (tautology) by F (contradiction)
-
Replacing F (contradiction) by T (tautology)
while keeping negations unchanged.
