CSIR NET MATHEMATICS NOV 2020 | 30th Nov | Section C | Real Analysis | Injective Function

Hello friends, this is Mahesh Ullekar and you are watching Next Gen Maths. If you have not subscribed my channel, then please subscribe the channel and press the bell icon. In this video, we are going to discuss the problem of CSI NIT, November 2020, problem of real analysis. Equation, let n equal to 1, 2, 3 with a set of natural numbers, each of the following function from n cross n to n are injective. We have given four options. First option, f1 equal to 2 power m, 3 power m. f2 equal to mn plus m plus n. f3 equal to m square plus n cube. f4 equal to m square into n cube. If you want notes of this solution, then please visit our website www.nextgenmaths.com. In the description, link is available. It is a problem of injective function, so we begin with the definition of injective function. A function f from a to b is said to be injective or one-to-one if f of a equal to f of b implies a equal to b. That means image of two numbers are equal, then the number must be equal or if a not equal to b, then f of a not equal to f of b. That means if numbers are not equal, then image must not be equal. Any function which satisfies this condition will be the injective function. We have given four options and we will check all the options one by one. So first option, f1mn equal to 2 power m, 3 power m. Suppose f1m1n1 equal to f1m2n2. That means image of two numbers are same. By the definition, it is 2 power m1, 3 power n1 equal to 2 power m2, 3 power n2. If we simplify, then we have 2 power m1 divided by 2 power m2 equal to 3 power n2 divided by 3 power n1. Now it is 2 power m1 minus m2 equal to 3 power n2 minus n1. Left hand side is a power of 2, so it will give the even number and right hand side is a power of 3, so it will give the odd number. They are not equal except m1 minus m2 equal to 0 and n2 minus n1 equal to 0. That means m1 equal to m2 and n2 equal to n1, which will give m1n1 equal to m2n2. Here we begin with that image of m1n1 and m2n2 are equal and here we show that numbers are also equal, that means m1n1 equal to m2n2. So this is a 1-1 function or you can say injective function. Therefore the option number 1 is the correct option. Now we are going to check second option f2mn equal to mn plus m plus n. Now it is a symmetric function in m and n, that means f2mn equal to f2nm. Therefore, f212 and f221 have the same image which is 5, but 12 not equal to 21. That means we have two numbers which are not equal, but their images are equal, therefore it is not an injective function. So option number 2 is the wrong answer. Now option number 3 and 4, we will check both the options at the same time. Option 4 has function f4mn equal to m square into n cube. We begin with the assumption f4m1n1 equal to f4m2n2. Therefore we have m1 square into n1 cube equal to m2 square into n2 cube. If we rewrite then m1 square divided by m2 square equal to n2 cube divided by n1 cube. Then you can rewrite in the form of m1 divided by m2 square equal to n2 divided by n1 cube. That means if we are able to find two numbers in which square of one equal to cube of another. And we have example 8 square equal to 4 cube. 8 square equal to 64 and 4 cube is also 64. Therefore m1 equal to 8m2 and n2 equal to 4n1. Now if we take m2 equal to 1, n1 equal to 1, then m1 equal to 8 and n2 equal to 4. Therefore m1n1 equal to 81 and m2n2 equal to 14. So image of f4 81 equal to 64 and image of f4 14 is also 64. So we are able to find two numbers which are not equal but their images are equal. So it also violate the condition of injective function. Therefore option number 4 will not give the injective function. So option number 4 is the wrong answer. Now the same numbers will work for the option number 3. So f3mn which is m square plus n cube. If we take image of f3 81 and f3 14 will we give the 65. But 81 not equal to 14. So f3 is not an injective function. Therefore option number 3 and option number 4 are the wrong option. Therefore there is only one correct option f1mn equal to 2 power m, 3 power n. Thanks for watching. For more videos like, share, subscribe, press the bell icon of NextGenMath.