#### Topics

##### Sets

- Sets and Their Representations
- Empty Set (Null or Void Set)
- Finite and Infinite Sets
- Equal Sets
- Subsets
- Power Set
- Universal Set
- Venn Diagrams
- Intrdouction of Operations on Sets
- Union of Sets
- Intersection of Sets
- Difference of Sets
- Complement of a Set
- Practical Problems on Union and Intersection of Two Sets
- Proper and Improper Subset
- Open and Close Intervals
- Disjoint Sets
- Element Count Set

##### Mathematical Reasoning

- Mathematically Acceptable Statements
- New Statements from Old
- Special Words Or Phrases
- Contrapositive and Converse
- Introduction of Validating Statements
- Validation by Contradiction
- Difference Between Contradiction, Converse and Contrapositive
- Consolidating the Understanding

##### Sets and Functions

##### Relations and Functions

- Cartesian Product of Sets
- Concept of Relation
- Concept of Functions
- Some Functions and Their Graphs
- Algebra of Real Functions
- Ordered Pairs
- Equality of Ordered Pairs
- Pictorial Diagrams
- Graph of Function
- Pictorial Representation of a Function
- Exponential Function
- Logarithmic Functions
- Brief Review of Cartesian System of Rectanglar Co-ordinates

##### Algebra

##### Trigonometric Functions

- Concept of Angle
- Introduction of Trigonometric Functions
- Signs of Trigonometric Functions
- Domain and Range of Trigonometric Functions
- Trigonometric Functions of Sum and Difference of Two Angles
- Trigonometric Equations
- Trigonometric Functions
- Truth of the Identity
- Negative Function Or Trigonometric Functions of Negative Angles
- 90 Degree Plusminus X Function
- Conversion from One Measure to Another
- 180 Degree Plusminus X Function
- 2X Function
- 3X Function
- Expressing Sin (X±Y) and Cos (X±Y) in Terms of Sinx, Siny, Cosx and Cosy and Their Simple Applications
- Graphs of Trigonometric Functions
- Transformation Formulae
- Values of Trigonometric Functions at Multiples and Submultiples of an Angle
- Sine and Cosine Formulae and Their Applications

##### Coordinate Geometry

##### Calculus

##### Complex Numbers and Quadratic Equations

- Concept of Complex Numbers
- Algebraic Operations of Complex Numbers
- The Modulus and the Conjugate of a Complex Number
- Argand Plane and Polar Representation
- Quadratic Equations
- Algebra of Complex Numbers - Equality
- Algebraic Properties of Complex Numbers
- Need for Complex Numbers
- Square Root of a Complex Number

##### Mathematical Reasoning

##### Linear Inequalities

##### Statistics and Probability

##### Permutations and Combinations

- Fundamental Principles of Counting
- Permutations
- Combination
- Introduction of Permutations and Combinations
- Permutation Formula to Rescue and Type of Permutation
- Smaller Set from Bigger Set
- Derivation of Formulae and Their Connections
- Simple Applications of Permutations and Combinations
- Factorial N (N!) Permutations and Combinations

##### Principle of Mathematical Induction

##### Binomial Theorem

- Introduction of Binomial Theorem
- Binomial Theorem for Positive Integral Indices
- General and Middle Terms
- Proof of Binomial Therom by Pattern
- Proof of Binomial Therom by Combination
- Rth Term from End
- Simple Applications of Binomial Theorem

##### Sequence and Series

##### Straight Lines

- Slope of a Line
- Various Forms of the Equation of a Line
- General Equation of a Line
- Distance of a Point from a Line
- Brief Recall of Two Dimensional Geometry from Earlier Classes
- Shifting of Origin
- Equation of Family of Lines Passing Through the Point of Intersection of Two Lines

##### Conic Sections

- Sections of a Cone
- Concept of Circle
- Introduction of Parabola
- Standard Equations of Parabola
- Latus Rectum
- Introduction of Ellipse
- Relationship Between Semi-major Axis, Semi-minor Axis and the Distance of the Focus from the Centre of the Ellipse
- Special Cases of an Ellipse
- Eccentricity
- Standard Equations of an Ellipse
- Latus Rectum
- Introduction of Hyperbola
- Eccentricity
- Standard Equation of Hyperbola
- Latus Rectum
- Standard Equation of a Circle

##### Introduction to Three-dimensional Geometry

##### Limits and Derivatives

- Intuitive Idea of Derivatives
- Introduction of Limits
- Introduction to Calculus
- Algebra of Limits
- Limits of Polynomials and Rational Functions
- Limits of Trigonometric Functions
- Introduction of Derivatives
- Algebra of Derivative of Functions
- Derivative of Polynomials and Trigonometric Functions
- Derivative Introduced as Rate of Change Both as that of Distance Function and Geometrically
- Limits of Logarithmic Functions
- Limits of Exponential Functions
- Derivative of Slope of Tangent of the Curve
- Theorem for Any Positive Integer n
- Graphical Interpretation of Derivative
- Derive Derivation of x^n

##### Statistics

- Measures of Dispersion
- Concept of Range
- Mean Deviation
- Introduction of Variance and Standard Deviation
- Standard Deviation
- Standard Deviation of a Discrete Frequency Distribution
- Standard Deviation of a Continuous Frequency Distribution
- Shortcut Method to Find Variance and Standard Deviation
- Introduction of Analysis of Frequency Distributions
- Comparison of Two Frequency Distributions with Same Mean
- Statistics Concept
- Central Tendency - Mean
- Central Tendency - Median
- Concept of Mode
- Measures of Dispersion - Quartile Deviation
- Standard Deviation - by Short Cut Method

##### Probability

- Random Experiments
- Introduction of Event
- Occurrence of an Event
- Types of Events
- Algebra of Events
- Exhaustive Events
- Mutually Exclusive Events
- Axiomatic Approach to Probability
- Probability of 'Not', 'And' and 'Or' Events

- Subsets of set of real numbers
- Intervals as subsets of R

## Definition

A 'set A' is said to be a subset of a set B if every element of A is also an element of B.

## Notes

Sub means 'part of', thus subset means part of set.

B= {1, 2, 3, 4, 5} and A= {1, 2}

A set is the part of set B

A is subset of B

Mathematically it is written as A ⊂ B

'⊂' means is subset of

B= {1, 2, 3, 4, 5}, A= {1, 2}, C= {3, 4}, D= {3, 5} and E= {1,5}

A, C, D and E are subsets of B

Mathematically, if A ⊂ B, a ∈ A ⇒ a ∈ B

'a' is an element of set A

'⇒' means implies

If A is not the subset of B, then mathematically we will write A ⊄ B

'⊄' means not the subset of

Case 1- If every elemento of A is also an element of B, i.e A ⊂ B

But if every element of B is also an element of A, i.e B ⊂ A

then A=B

A ⊂ B and B ⊂ A ⇔ A= B

'⇔' means two way implication

Note: 1) A = A

A⊂ A. Every set is subset of itself.

2) Ø ⊂ any set . Empty set is subset of any set.

**There are many important subsets of R (set of all real numbers). We give below the names of some of these subsets. **

The set of natural numbers N = {1, 2, 3, 4, 5, . . .}

The set of integers Z = {. . ., –3, –2, –1, 0, 1, 2, 3, . . .}

The set of rational numbers Q = { x : x = p/q, p, q ∈ Z and q ≠ 0}

The set of irrational numbers, denoted by T, is composed of all other real numbers. Thus T = {x : x ∈ R and x ∉ Q}, i.e., all real numbers that are not rational.

Members of T include 2 , 5 and π .

Some of the obvious relations among these subsets are:

N ⊂ Z ⊂ Q, Q ⊂ R, T ⊂ R, N ⊄ T.

Let A & B be two sets. A ⊂ B and A ≠ B.

Then A is called proper subset. and B is called **superset**.

A set containing only one element is known as **Singleton set**.

A= {1}, B= {Ø}, C= {2}

**Intervals as subset of R**

If A= {x: 5 ≤ x ≤ 9 ∀ x ∈ R}

Real Interval [5,9] is subset of R, here 5 and 9 are included therefore we wrote them in these '[ ]' brackets and such intervals are known as closed intervals.

If B= {x:5 < x < 9, ∀ x ∈ R}

Real Interal (5,9) is subset of R, here 5 and 9 are excluded therefore we wrote them in these '( )' brakets and such intervals are known as open intervals.

If C= {x: 5 < x ≤ 9 ∀ x ∈ R}

Real interval (5,9] is subset of R, here 5 is excluded and 9 is included therefore we wrote them in '(' and ']' brackets such intervals are known as open closed intervals.

If D= {x: 5 ≤ x < 9, ∀ x ∈ R}

Real interval [5,9) is subset of R, here 5 is included and 9 is excluded therefore we wrote them in '[' and ')' brackets such intervals are known as closed open intervals.

**Length of Interval:** if [a,b], (a,b), [a,b), (a,b] are the subsets, then b-a is the length of interval.