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Karnataka Class 11 Commerce Maths UNIT I SETS AND FUNCTIONS

Karnataka Class 11 Commerce Maths UNIT I SETS AND FUNCTIONS

Karnataka Class 11 Commerce Maths UNIT I SETS AND FUNCTIONS

Karnataka Class 11 Commerce Maths UNIT I SETS AND FUNCTIONS : The theory of sets was developed by German Mathematician George Cantor. He is regarded as the father of set theory. It is proved to be of great importance in the foundation of relations and functions, sequences, Geometry, Probability theory etc. Also it has wide application in logic and philosophy.

Karnataka Class 11 Commerce Maths UNIT I SETS AND FUNCTIONS – syllabus

UNIT I: SETS AND FUNCTIONS
1. Sets

  • Sets and their representations:
  • Definitions, examples, Methods of Representation in roster and rule form, examples
  • Types of sets: Empty set. Finite and Infinite sets. Equal sets. Subsets.
  • Subsets of the set of real numbers especially intervals (with notations).
  • Power set.
  • Universal set. examples
  • Operation on sets: Union and intersection of sets. Difference of sets. Complement of a set,
  • Properties of Complement sets. Simple practical problems on union and intersection of two sets.
  • Venn diagrams: simple problems on Venn diagram representation of operation on sets

2. Relations and Functions

  • Cartesian product of sets: Ordered pairs, Cartesian product of sets.
  • Number of elements in the Cartesian product of two finite sets.
  • Cartesian product of the reals with itself (upto R × R × R).
  • Relation: Definition of relation, pictorial diagrams, domain, co-domain and range of a relation and examples
  • Function : Function as a special kind of relation from one set to another.
  • Pictorial representation of a function, domain, co-domain and range of a function.
  • Real valued function of the real variable, domain and range of constant, identity, polynomial rational, modulus, signum and greatest integer functions with their graphs.
  • Algebra of real valued functions:
    Sum, difference, product and quotients of functions with examples.

3. Trigonometric Functions

  • Angle: Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another.
  • Definition of trigonometric functions with the help of unit circle. Truth of the identity sin2x + cos2x = 1, for all x.
  • Signs of trigonometric functions and sketch of their graphs.
  • Trigonometric functions of sum and difference of two angles.
  • Definition of allied angles and obtaining their trigonometric ratios using compound angle formulae.
  • Trigonometric ratios of multiple angles:
  • Identities related to sin2x, cos2x, tan2x, sin3x, cos3x and tan3x.
  • Trigonometric Equations:
  • General solution of trigonometric equations of the type sinθ = sin α, cosθ = cosα and tanθ = tan α. and problems.
  • Proofs and simple applications of sine and cosine rule.

Karnataka Class 11 Commerce Maths UNIT I SETS AND FUNCTIONS

Introduction:

The theory of sets was developed by German Mathematician George Cantor. He is regarded as the father of set theory. It is proved to be of great importance in the foundation of relations and functions, sequences, Geometry, Probability theory etc. Also it has wide application in logic and philosophy.

Karnataka Class 11 Commerce Maths UNIT I SETS AND FUNCTIONS

Sets:

A set is a well defined collection of distinct objects. Each member is called the element of the set.

Note:

1. A set is always represented by capital letters

2. If a is an element of set A then we write a ∈ A.

3. If b is not an element of set A then we write b ∉ A

Examples:

1. The set of boys in class VthA.

2. The set of even natural numbers

3. The set of days of a week

4. The set of vowels in the English alphabet.

Karnataka Class 11 Commerce Maths UNIT I SETS AND FUNCTIONS

Methods of describing a set:

A set can be represented in two forms
1. Roster form or Tabular form
2. Set builder form or rule form

Roster Form:

In the roster Form, all the elements are listed and separated by commas and are enclosed within brackets.

A = The set of all even numbers between 0 and 10

Roster Method given by A = {2, 4, 6, 8}

Set builder Form:

In this method all the elements of a set possess a single common property, which is not possessed by any element outside the set.
If A = {1, 2, 3, 4, 5} then the set builder form is represented by A = { x : x N and x < 6}

Karnataka Class 11 Commerce Maths UNIT I SETS AND FUNCTIONS

Null set or Empty set:

A set containing no elements is called an empty set.
It is denoted by Φ or { }
For eg.: A = {The set of all even prime numbers other than 2 }
A = Φ or { }
2. A = set of all natural numbers < 0
A = Φ  or{ }

Singleton set:

A set containing only one element is called a singleton set.
Eg. 1. A = { x : x – 1 = 0 , x ∈ N}
A = { 1 }
2. B = {x : x is an even prime number}
B = { 2 }

Finite set and infinite set:

A set is called a finite set if it contains finite numbers of elements.

Example

1) A = {1, 2, 3}
n(A) = 3

2) B = {set of prime numbers < 9}
B = {2, 3, 5,7}
n(B) = 4
A set which is not finite is called an infinite set.

Examples

1. The set of natural numbers.
2. The set of real numbers.

Karnataka Class 11 Commerce Maths UNIT I SETS AND FUNCTIONS

Equal and Equivalent sets:

Equal Sets: Two sets A and B are said to be equal if they have exactly the same elements.

Ex.

1). A = {1, 3, 8}
B = {8, 3, 1}
Then A = B as A and B have the same elements.

2). A = { x : x is a letter in the word ‘flow’}
B = { x : x is a letter in the word ‘wolf’)
Then A = B as A and B have the same elements.

Karnataka Class 11 Commerce Maths UNIT I SETS AND FUNCTIONS

Equivalent sets:

Two finite sets A and B are said to be equivalent if they have the same cardinal number i.e. if the same number of elements. i.e. if
n (A) = n (B).
Let A = {a, e, i, o, u}
B = {1, 2, 3, 4, 5}
Then n (A) = 5 and n (B) = 5
⇒ the sets A and B are equivalent.

Subset:

If each and every element of A is an element of B, then A is called a subset of B or A is contained in B. We write A  B.

Example  1

A = {1, 2}
B = {1, 2, 3, 4}
A ⊂ B

Note:

1. If atleast one element of A does not belong to set B then A is not a subset of B. It is symbolically represented by A⊆B

2. Every set A is a subset of itself i.e. A⊆A.

3. Φ is a subset of every set.

4. If A⊆B and B⊆ A then A = B

Example 2

Set of Natural Numbers ⊆ set of whole numbers

Super Set:

Set A and B are two non empty sets such that A is contained in B and A ≠ B then B is called the super set of A

It is symbolically represented by B ⊃A

EXAMPLE : Set of complex numbers is a super set of set of real numbers

Karnataka Class 11 Commerce Maths UNIT I SETS AND FUNCTIONS

PROPER SUBSET:

A is called a proper subset of B if each and every element of A is contained in B and A  B. It is symbolically represented by A ⊂ B and is read as ‘A’ is a proper subset of B
A = {a, b, c}
B = {a, b, c, d}
A ⊂ B

Power Set :

A set formed by all the subsets of a set A as its elements is called the Power set of A and is denoted by p(A).

Universal set:

If all the sets under consideration are subsets of a set U. Then U is called the Universal set.

Example:

For the set of integers Z, the universal set can be set of real numbers
R or the set of complex numbers C.

Cardinal number of a finite set:

The number of elements of a finite set A is called the cardinal number and is represented by n(A).
A = {1, 2, 3, 4, 5, 6}
Cardinal number of set A = n (A) = 6.

Karnataka Class 11 Commerce Maths UNIT I SETS AND FUNCTIONS

Operation on Sets:

a) Union of Sets:

Let A and B be any two sets. Then the union of A and B denoted by A∪ B is defined to be the set of all those elements. which are in A or in B or in both.

Examples:

1. Let A = {a, b, c} B = {c, d, e, f}
∴ A∪ B = {a, b, c, d, e, f}

2. A = {1, 2, 3, 4, 5}
B = { 1, 2, …………9}
A∪ B = {1, 2, 3 …… ..9}

Note:

1) A ∪ A = A
2) A∪ Ø = A
3) Ø∪ Ø = Ø
4) If A ⊆ B then A∪ B = B

b) Intersection of sets:

Let A and B be any two sets. Then the intersection of A and B denoted by A ∩ B is defined to be the set of all common elements between A and B

Example:

1. Let A = {a, b, c, d}
B = {c, d, e, f, g, h}
A ∩ B = {c,d}

2. Let A = {1, 2, 3, 4, 5}
B = {1, 2, 3, ….. 9}
A ∩ B = {1, 2, 3, 4, 5} = A itself

Note:

1. A ∩ A = A
2. A ∩ Ø = Ø
3 Ø ∩ Ø = Ø
4. It A ⊆ B then A ∩ B = A itself.

c) Difference between any two sets :

Let A and B be any two sets. Then the difference A-B is defined to be the set of all those elements of A which are not in B.

It is also called the complement of B w.r.t. A. Similarly B – A is defined to be the set of all those elements of B which are not in A. It is also called the complement of A w.r.t. B.

Example:

1. Let A = {a, b, c, d}
B = {d, e, f, g, h, i}
A – B = {a, b, c}
B – A = {e, f, g, h, i}

Function is a special type of relation. Each function is a relation but each relation is not a function. In this lesson we shall discuss some basic definitions and operations involving sets, Cartesian product of two sets, relation between two sets, the conditions under when a relation becomes a function, different types of function and their properties.

Karnataka Class 11 Commerce Maths UNIT I SETS AND FUNCTIONS

Relations and Functions

Cartesian product of sets: Ordered pairs, Cartesian product of sets. Number of elements in the Cartesian product of two finite sets. Cartesian product of the reals with itself (upto R × R × R).

Relation: Definition of relation, pictorial diagrams, domain, co-domain and range of a relation and examples

Function : Function as a special kind of relation from one set to another. Pictorial representation of a function, domain, co-domain and range of a function. Real valued function of the real variable, domain and range of constant, identity, polynomial rational, modulus, signum and greatest integer functions with their graphs.

Algebra of real valued functions: Sum, difference, product and quotients of functions with examples.

Karnataka Class 11 Commerce Maths UNIT I SETS AND FUNCTIONS

Functions:

Set X and Y be two non-empty sets A subset f of X×Y is called as a function if the following conditions hold good.

i) For each x ∈ X , there exists a unique y ∈ Y such that (x, y) ∈ f ii) Elements of x should not be repeated.

Note: 1) If (x1 , y1 ) ∈f and (x2, y2 ) ∈ f then y1 = y2

2) y ∈Y is called as the image of the element x ∈ X

3) x is called as the pre-image of y ∈ Y.

4) The set X is called as the Domain of the function f.

5) The set Y is called as the Range of the function f.

Domain, Co-domain and Range of a function:

Let F : A B i.e. F is a function or mapping from A to B. The set A is called as domain of F and the set B is called as Co-domain of F.

The set consisting of all images is called as the range of F which in symbols is written on F(A)F = { (1, a), (2, b), (3, c), (4, d)}

Domain of F = A= {1, 2, 3, 4} Range of F =F(A)= {a, b, c, d}

Co-domain of F = B= {a, b, c, d)

Note: 1) F (A) ⊆ B

2) Every function is a relation but the converse is not true.

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