TLDR;
This video provides a comprehensive one-shot explanation of Relations and Functions for Class 11 Maths, focusing on building a strong foundation from basic concepts to advanced topics. It emphasizes understanding the prerequisite of Sets, the importance of patience, and the practical application of the chapter in both Class 11 and 12 Maths. The lecture covers ordered pairs, Cartesian products, relations, functions, domains, ranges, and various types of functions with their graphical representations.
- Ordered pairs and Cartesian products are fundamental to understanding relations and functions.
- Relations are subsets of Cartesian products, linking elements between sets based on specific conditions.
- Functions are special types of relations where each element of the domain maps to exactly one element in the range.
- Domain, range, and codomain are essential for defining and analyzing functions.
- Real-valued functions and their graphs are crucial for visualizing and understanding functional relationships.
Introduction [0:00]
The video aims to comprehensively cover relations and functions, assuming prior knowledge of sets. It promises to explain the concepts in a way that enables viewers to teach others. The content is designed to build a strong foundation for use in both 11th and 12th grade math. Patience is advised due to the length of the video, with the suggestion to watch it in segments if needed.
Ordered pairs [2:40]
An ordered pair consists of two objects or elements written in a specific order. The order is important, meaning (1, 2) and (2, 1) are different. In geometry, ordered pairs indicate the location of a point. The first element is 'a' and the second element is 'b' in the ordered pair (a, b). An ordered pair is not a set, but a set is a collection of objects.
Cartesian product of sets [5:32]
The Cartesian product of two sets, A and B, is a set of all possible ordered pairs where the first element comes from set A and the second element comes from set B. For example, if A = {1, 2} and B = {3, 4}, then A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}. The order matters, so A × B is generally not equal to B × A. The Cartesian product can be represented in roster form, listing all ordered pairs, or in set builder form, describing the property that defines the ordered pairs.
Number of elements in the cartesian product of two sets [11:35]
If set A has 'm' elements and set B has 'n' elements, then the Cartesian product A × B will have m * n elements. This is a fundamental formula for determining the size of a Cartesian product. If two ordered pairs are equal, their corresponding elements must be equal. The video then solves an example problem to find x and y given two equal ordered pairs.
Relation [42:05]
A relation is a connection or link between two sets. Mathematically, a relation is a subset of the Cartesian product of two sets. This means a relation is also a set containing ordered pairs. For example, if A = {Kohli, Vicky, Safe} and B = {Anushka, Katrina, Kareena}, a relation "is the husband of" would result in the set {(Kohli, Anushka), (Vicky, Katrina)}.
Total number of relations [51:38]
The total number of relations that can be defined from a set A to a set B is equal to the number of subsets of A × B. If A × B has 'n' elements, then there are 2^n possible relations. A relation is defined on the set of natural numbers, and an example is given to write the relation in roster form, where the second element is the cube of the first element, with the first element being a prime number less than 10.
Representation of a relation [1:03:12]
A relation, being a set, can be represented in roster form (listing the ordered pairs), set builder form (defining the property that connects the elements), or using an arrow diagram.
Domain and Range of relation [1:03:56]
The domain of a relation is the set of all first elements of the ordered pairs in the relation. The range is the set of all second elements. The codomain is the entire set B to which the second elements belong. The video uses an arrow diagram to illustrate these concepts, also defining "image" and "pre-image." The domain consists of elements in the first set that have an image in the second set, while the range consists of elements in the second set that have a pre-image in the first set.
Relation on a set [1:14:33]
A relation on a set A is a relation from A to itself, meaning it is a subset of A × A. The video provides an example and asks to find the domain and range of the relation.
Inverse of a relation [1:20:15]
The inverse of a relation R is obtained by swapping the elements in each ordered pair. If R = {(a, b)}, then R⁻¹ = {(b, a)}. The domain of R is the range of R⁻¹, and the range of R is the domain of R⁻¹. The video provides an example and explains how to find the inverse relation, its domain, and its range.
What is a function? [1:50:57]
A function is a special type of relation where every element of set A has an image in set B, and no element of set A has more than one image. The video uses arrow diagrams to illustrate when a relation is a function and when it is not. Converging is allowed, but diverging is not.
Description of a function [2:26:47]
If 'f' is a function from A to B, then B is called the image of A under f. The video revisits the definition of a function and uses arrow diagrams to determine whether given relations are functions.
Equal function [2:46:19]
Two functions, f and g, are equal if they have the same domain, the same codomain, and f(x) = g(x) for all x in their domain. The video provides an example to determine whether two given functions are equal.
Real valued function [2:54:47]
A real-valued function is a function whose range is a subset of the real numbers. If both the domain and range are subsets of real numbers, it is called a real function.
Domain [3:02:01]
The domain of a function is the set of all possible input values (x) for which the function is defined. The video provides examples of finding the domain of various functions, considering restrictions such as division by zero and square roots of negative numbers.
Range of a function [3:19:22]
The range of a function is the set of all possible output values (f(x)) that the function can produce. The video explains how to find the range, often by expressing x in terms of y and then determining the possible values of y.
Some standard real functions and their graphs [3:49:27]
The video discusses several standard real functions and their graphs, including constant functions, identity functions, reciprocal functions, square functions, modulus functions, signum functions, and greatest integer functions. For each function, the domain, range, and key characteristics of the graph are explained.
Thank You Bacchon [4:21:24]
The video concludes with a thank you message.