Use the Expertise provided here to learn Relations and Functions. Relations and Functions are the stepping stones to understanding Calculus. You'll start with the very basic concept of a Relation.
You can visit here for free resources on Relations and Functions. All the formula that you need to know in Relations and Functions for class 12 Mathematics NCERT, ISC and the CUET exams are given here in this page. All you need to do is just scroll down the page!
You will learn about the Identity Relation, Empty Relation and Universal Relation. The 3 types of Relations, namely Reflexive , Symmetric and Transitive Relations are explained here. Note that a Relation is said to be an Equivalence Relation, if it is Reflexive, Symmetric and Transitive.
What are the types of Relations?
There are 3 types of Relations, namely, Reflexive, Symmetric and Transitive Relation and the following posts help explain this.
A Relation is said to be an equivalence Relation if it is Reflexive, Symmetric and Transitive. In this context, you will also be introduced to equivalence classes.
Consequently, you will proceed to learn about functions.
A function f from X to Y is a relation which assigns to each element in X, a unique element in Y. If f(x)= y, y is called the image of x, x is the preimage of y.
In this context, we have the vertical line test, which states that if a vertical line meets the graph of a function at exactly 1 point, then it is a function.
You will now learn about the domain and range of a function and most importantly, the difference between the codomain and range.
You will now be introduced to a few important and standard functions, namely, the constant function, identity function, modulus function and the greatest integer function along with the signum function. These are functions which every student of Calculus should know.
You will now learn about the types of functions,
One-one or injective function-A function f from X to Y is said to be injective, if distinct elements in X are mapped to distinct elements in Y. There is a small test to check if a function is injective.
In this context, we have the horizontal line test. A function is said to be one-one if a horizontal line meets the graph of a function at exactly 1 point.
A function f from X to Y is onto, if every element in Y has a pre-image in X.
A function f from X to Y is said to me many-one, if more than one element in X is mapped to one image in Y.
A function f from X to Y, is said to be into, if there exists at least one element in Y which does not have a pre-image in X.
What is a bijective function?
A function which is both injective and surjective is called a bijective function.
You are now going to learn 3 very useful formulas which are little known but immensely important, especially in multiple choice questions.
If A has m elements, B has n elements, then the number of one-one functions from A to B and the number of bijective functions from A to B is given here.
How about the number of onto functions from A to B?
The following post helps explain this! Normally, you may find this formula a little difficult to memorise. Fear not!
There are particular cases for this formula for n = 2 and
n = 3. Read this image to learn! This is a valuable tip for you, so please note!
The next most important concept that you'll need to know is about Invertible functions. Keep in mind that a function is invertible iff it is bijective. An important property that you'll need to keep in mind is that inverse of a function is unique.
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In this context, you will need to know what are composite functions and the domain and range of composite functions.
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And finally you'll conclude with Binary Operations. So, the following image shows you what is a Binary Operation.
