Bihar Board - Class 12 Math - Chapter 1: Relations and Functions Handwritten Notes

In mathematics, relations and functions are fundamental concepts that deal with how elements from one set relate to elements of another set. A relation is a connection or association between elements of two sets, whereas a function is a special type of relation where every element in the domain is associated with exactly one element in the codomain. This chapter explores the definitions, types, and properties of relations and functions, which are essential for understanding many concepts in algebra and calculus.

Key Points:

1. Relations:

   • A relation is a set of ordered pairs, where each pair contains an element from the first set (domain) and an element from the second set (codomain).
   • Example: If $$A={1,2,3}A = \{1, 2, 3\} and $$B={a,b,c}B = \{a, b, c\}, a relation from $$AA to $$BB can be $$R={(1,a),(2,b),(3,c)}R = \{(1, a), (2, b), (3, c)\}.
   • Relations can be represented using sets of ordered pairs, matrices, or directed graphs.

2. Types of Relations:

   • Reflexive Relation: Every element is related to itself. For set $$A={1,2,3}A = \{1, 2, 3\}, the relation $$R={(1,1),(2,2),(3,3)}R = \{(1, 1), (2, 2), (3, 3)\} is reflexive.
   • Symmetric Relation: If an element $$aa is related to $$bb, then $$bb is related to $$aa as well. Example: $$R={(1,2),(2,1)}R = \{(1, 2), (2, 1)\}.
   • Transitive Relation: If $$aa is related to $$bb and $$bb is related to $$cc, then $$aa must be related to $$cc. Example: $$R={(1,2),(2,3),(1,3)}R = \{(1, 2), (2, 3), (1, 3)\}.
   • Anti-symmetric Relation: If $$aa is related to $$bb and $$bb is related to $$aa, then $$a=ba = b.

3. Functions:

   • A function is a specific type of relation where each element of the domain is associated with exactly one element of the codomain.
   • Example: $$f:A→Bf: A \to B, where for every $$x∈Ax \in A, there is exactly one $$y∈By \in B.
   • Functions can be represented by graphs, tables, or equations.

4. Types of Functions:

   • One-to-One (Injective): Every element of the domain is mapped to a unique element in the codomain.
   • Onto (Surjective): Every element of the codomain has at least one element from the domain that maps to it.
   • One-to-One Correspondence (Bijective): A function that is both injective and surjective.
   • Constant Function: A function where all elements in the domain map to the same element in the codomain.
   • Identity Function: A function where each element of the domain is mapped to itself.

5. Domain and Range:

   • The domain of a function is the set of all possible inputs, and the range is the set of all possible outputs.
   • Example: For the function $$f(x)=x2f(x) = x^2, the domain is all real numbers, and the range is all non-negative real numbers.

6. Operations on Functions:

   • Addition, Subtraction, Multiplication, and Division of functions can be performed by applying these operations to their individual expressions.
   • Composition of Functions: If $$f:A→Bf: A \to B and $$g:B→Cg: B \to C, then the composition $$g∘fg \circ f is a function from $$AA to $$CC, defined by $$(g∘f)(x)=g(f(x))(g \circ f)(x) = g(f(x)).

Conclusion:

Relations and functions are crucial concepts in mathematics that help establish connections between sets and provide the foundation for more advanced topics like calculus and algebra. Understanding the various types of relations (such as reflexive, symmetric, and transitive) and functions (including injective, surjective, and bijective functions) allows for deeper insights into mathematical structures and real-world applications. These concepts are essential for solving problems in a wide range of mathematical and scientific fields.