Relations and Functions is one of the most important chapters in the Class 12 NCERT syllabus for mathematics, covering essential concepts that form the foundation for higher mathematics and competitive exams like JEE. Whether you're preparing for CBSE Class 12 board exams or JEE 2024, mastering this chapter is key to your success.
In this article, we’ll cover everything from the basics to advanced topics like different types of relations, functions, and binary operations, along with important NCERT examples and exercises.
Introduction to Relations and Functions
A relation in mathematics connects elements from two sets, while a function is a special type of relation where every element in one set is associated with exactly one element in another set. Understanding these concepts is crucial for tackling problems in algebra, calculus, and geometry.
Types of Relations
Relations are categorized into various types based on specific properties:
1. Empty Relation:
2. Universal Relation:
3. Reflexive Relation:
4. Symmetric Relation:
5. Transitive Relation:
6. Equivalence Relation:
Example Problems:
- NCERT Ex 1.1 Q1 (i)
- NCERT Ex 1.1 Q12
Types of Functions
Functions describe a relationship between two sets where each element in the domain has one and only one corresponding element in the co-domain. The different types of functions include:
1. One-One (Injective) Function: Every element in the domain maps to a unique element in the co-domain.
2. Many-One Function: Multiple elements in the domain may map to the same element in the co-domain.
3. Onto (Surjective) Function: Every element in the co-domain has at least one corresponding element in the domain.
4. One-One and Onto (Bijective) Function: A function that is both injective and surjective.
Important NCERT Examples:
- NCERT Example 8
- NCERT Ex 1.2 Q2 (i), (ii)
- NCERT Ex 1.2 Q6
Composition of Functions
Composition of functions involves applying one function to the result of another. This concept is critical in higher-level mathematics and has multiple applications.
Practice Problems:
- NCERT Ex 1.3 Q1
- NCERT Ex 1.3 Q5
Invertible Functions
A function is invertible if there exists another function that undoes its action. This inverse function maps each element of the co-domain back to its original element in the domain.
Binary Operations
Binary operations combine two elements of a set to produce another element from the same set. Key properties include:
- Commutative Property: The operation gives the same result regardless of the order of operands.
- Associative Property: The grouping of operands does not change the result of the operation.
- Identity Element: An element that does not change other elements when used in an operation.
- Inverse Element: An element that, when combined with another, results in the identity element.
Example Problems:
- NCERT Ex 1.4 Q1 (i), (ii), (iii)
- NCERT Ex 1.4 Q2 (i)
JEE 2024: Importance of Relations and Functions
For JEE aspirants, understanding relations and functions is essential, as it is a topic frequently tested in both JEE Main and Advanced exams. Functions, in particular, are a foundational concept for calculus, which makes up a significant portion of the JEE syllabus.
Important Theorems and Tips for JEE:
- Focus on solving different types of function problems.
- Practice NCERT examples and exercises thoroughly.
- Pay special attention to binary operations and invertible functions.
Crash Course: Relations and Functions
For those looking to quickly revise, Deepak Sir's crash course video on Relations and Functions for Class 12 offers a complete overview of the chapter in a single video, including step-by-step NCERT solutions, important notes, and problem-solving techniques.
Watch Full video Here
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By mastering the concepts discussed in this article, students can confidently tackle questions on relations and functions in their board exams and competitive tests like JEE 2024. Make sure to practice NCERT examples and exercise problems, as these are often directly referenced in exams.
Watch the full video lecture and solution walkthrough by Deepak Sir to clear all your doubts and excel in your exams!