Relations and Functions Class-12th

Relations and Functions Class-12th: In Class 12 Mathematics, the chapter on Relations and Functions is foundational for understanding more complex concepts in higher mathematics. This chapter extends the basics learned in Class 11 and delves deeper into relations, types of functions, and compositions of functions, which are crucial for understanding calculus and other advanced topics.

Relations and Functions Class-12th Video Tutorial

Relations and Functions Class-12th

A relation in mathematics is a rule that connects elements of two sets, linking an element from one set (the domain) to one or more elements from another set (the codomain). If each element of the domain is linked to exactly one element of the codomain, the relation is called a function.

What is Sets?

A set is a well-defined collection of distinct objects, considered as an object in its own right. The objects or elements of a set can be anything: numbers, letters, symbols, or even other sets. Sets are a fundamental concept in mathematics and are used to describe collections of objects.

Elements of a Set:

The objects in a set are called its elements or members. If an element x belongs to a set A, we write x ∈ A. If x does not belong to set A, we write x ∉ A.

Representation of a set:

Sets are usually denoted by capital letters like A, B, C, etc. while their elements are written inside curly braces {}. There are two main ways to represent sets:
Roster or Tabular Form
Set-builder Form

Roster or Tabular Form: All elements of the set are listed, separated by commas, inside curly braces. For example,
A = {2,4,6,8,10}

Set-builder Form: A set is defined by a property that is members must satisfy. For example, the set of all even numbers can be written as:
A = { x | x is an even number}

Types of Sets:

Empty Set(Null Set): A set that contains no elements is called ‘empty set’ and it is denoted by Φ or {}.
Example:
The set of prime numbers between 4 and 6 is an empty set because there are no prime numbers between them.

Finite Set: A set with a limited number of elements.
Example:
A = {2,3,5,7,11,13} this is finite because set A have 6 elements.

Infinite Set: A set with an unlimited number of elements.
Example:
The set of natural numbers i.e., N = {1, 2, 3, 4, 5, . . .} is infinite.

Equal Sets: Two sets A and B are said to be equal if they contain exactly the same elements.
Example:
A = B if every element of A is in B, and every element of B is in A.

Subset: A set A is a subset of set B if every element of A is also an element of B. This is written as A ⊆ B.

Proper Subset: A set A is a Proper Subset of B if A ⊆ B and A ≠ B, meaning A contains some but not all elements of B.
This is written as A ⊂ B.

Universal Set: The universal set, denoted by U, is the set that contains all possible elements under consideration, typically for a particular problem.

Power Set: The power set of a set A is the set of all subsetss of A, including the empty set and A itself. It is denoted by P(A). If A has n elements, its power set contains 2^n subsets.

Sets Operations

Union: The union of two sets A and B is the set of all elements that belong to A, B, or both. It is denoted by A ∪ B.
Example:
if A = {1,2,3} and B = {3,4,5}, then:
A ∪ B = { 1,2,3,4,5}

Intersection: The intersection of two sets A and B is the set of all elements that are common to both A and B. It is denoted by A ∩ B.
Example: A = {2,3,4,5,6} and B = {1,2,3,8,9,10} then,
A ∩ B is { 2,3 }.

Difference: The difference of two sets A and B, denoted by A – B, is the set of elements that belong to A but not to B.
Example:
A = { 1,2,3,5 } and B = {2,3,4} then,
A – B is {1,5}

Venn Diagram

Venn diagrams are graphical representations of sets and their operations. In a Venn diagram, sets are represented by closed shapes (usually circles), and their relationships (like union, intersection, and difference) are illustrated by the overlapping or non-overlapping areas of these shapes.

Relations

A relation is defined as a relationship between the elements of two sets. If A and B are two non-empty sets, then a relation R from set A to set B is a subset of the Cartesian Product A x B. It is essentially a rule that associates an elements of one set to one or more elements of another set.

Cartesian Product:

The Cartesian Product of two sets A and B is defined as the set of all ordered pairs, where the first element belongs to A and the second belongs to B.
A x B = {(a,b) | a ∈ A and b ∈ B}
Example:
If A = {1,2} and B = [a3, 4}, the Cartesian Product A x B will be:
A x B = {(1,3),(1,4),(2,3),(2,4)}

Domain, Co-domain and Range:

Domain: The set of all possible first elements (input values) of the ordered pairs in a relation.
Codomain: The set in which the second element of each pair is supposed to lie (output values).
Range: The set of actual output values (second elements) from the relation.

Types of Relations

Reflexive Relation: A relation R on a set A is said to be reflexive if every element of A is related to itself.
for every a ∈ A, (a,a) ∈ R.
Example:
A = {1,2,3} then the relation R = {(1,1), (2,2), (3,3)} is reflexive.

Reflexive Relation: A relation R on a set A is said to be reflexive if every element of A is related to itself.
for every a ∈ A, (a,a) ∈ R.
Example:
A = {1,2,3} then the relation R = {(1,1), (2,2), (3,3)} is reflexive.

Symmetric Relation: A relation R on a set A is said to be symmetric if for every pair (a,b) ∈ R, (b,a) also belongs to R. In other words, if a is related to b, then b must also be related to a.

Transitive Relation: A relation R on a set A is transitive if for all a, b, c ∈ A, whenever (a,b) ∈ R and (b,c) ∈R then (a,c) ∈ R. Transitivity is an important property for many mathematically structures.

Equivalence Relation: A relation is called an equivalence relation if it is reflexive, symmetric, and transitive, Equivalence relations partition a set into equivalence classes where all elements are related to each other.

Functions

A function is defined as a set of ordered pairs (a,b), where each element a from the set A (the domain has exactly one corresponding element b in the set B (the codomain). This written as:
f : A → B, f(a) = b

A is called the domain, and B is called the co-domain. The set of all values of b, which correspond to elements in the domain, is called the range.

Types of Functions

One-to-One ( Injective ) Function:

A function f : A → B is said to be injective if different elements in A map to different elements in B. This means that no two elements in the domain are mapped to the same element in the co-domain.
Mathematically, if f(a1 ) = f(a2 ), then a1 = a2 .

Bijective Function:

A function is bijective if it is bothe injective and surjective. This means that every element in the domain maps to a unique element in the codomain, and every element in the co-domain has a corresponding element in the domain.
Bijective function are also called ‘One-to-One’ correspondence and have an inverse.

Constant Function:

A function is constant if all elements of the domain map to the same element in the co-domain.
Mathematically, if f : A → B, then f(a1 ) = f(a2 ) for all a1 , a2 ∈ A.

Graphical Representation of Functions

The graph of a function is a visual representation of the relationship between the domain and the codomain. For instance, the graph of a linear function is a straight line, while the graph of a quadratic function is a parabola. By analyzing graphs, we can easily understand the behavior of functions such as increasing, decreasing, or constant functions.

Algebra of Functions

The algebra of functions includes operations like:

Sum of Functions: If f : A → B and g : A → B , then the sum f + g : A → B is defined by (f + g)(x) = f(x) + g(x).
Difference of Functions: The difference f – g : A → B is defined by (f – g)(x) = f(x) – g(x).

Difference of Functions: The difference f – g : A → B is defined by (f – g)(x) = f(x) – g(x).
Product of Functions: The product f . g : A → B is defined by (f . g)(x) = f(x) . g(x).
Quotient of Functions: The quotient f / g : A → B is defined by (f / g)(x) = f(x) / g(x), provided that g(x) ≠ 0.

Composition of Functions

The composition of two functions is an operation where the output of one function becomes the input of another. If f : A → B and g : B → C, then the composition g∘f : A → C is defined as:
(g∘f)(x) = g(f(x))
Composition of functions is important in understanding how functions interact and combine in more complex mathematical processes.

Inverse of Functions

A function f : A → B has an inverse function f-1 : B → A if f(f-1 (b)) = b for all b ∈ B, and f-1(f(a)) = a for all a ∈ A. Only bijective functions have inverses because only these functions guarantee a one-to-one correspondence between elements of A and B.

Binary of Operations

A binary operation on a set A is a rule that assigns to each pair of elements in A a unique element in A. Common binary operations include addition, multiplication, and composition of functions.

Binary operations can be commutative (order does not affect the result) or associative (grouping of operations does not affect the result). These properties are important in algebraic structures like groups and rings.

Summary

The chapter covers the basics of how elements from sets relate through relations and how they map systematically using functions. Key concepts like injective, surjective, and bijective functions, along with binary operations, form the foundation for advanced mathematical studies and practical applications.

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