Question

Please give a response to below paragraph the one marked response not the question.

The Question:Discuss the properties of some of the commonly used functions on the set of Integers and on the set of Real Numbers. (Ex. Is the exponentiation any or none of Injective, Surjective, Bijective?, etc.)

Response: An exponential function such as exp(x) = e^x is an example of a commonly used injective function, which is a function that is one-to-one, meaning the elements of the domain are not mapped to the same codomain. The example, e^x is not surjective meaning, that the function does not have a right inverse (exp(x)^-1= e^x), in other words, every point in the codomain needs to be mapped to the domain to be considered a surjective function. At first, I was a little confused about what a bijection was as the definition sounds similar to an injective function, but if I understand it right, a bijection refers to a one-to-one correspondence. The difference being that a bijection is a function that is injective and surjective at the same time, and the exponential example (exp(x)=e^x) is not a bijective function since it is injective but not surjective.

Answer 1

First, let us understand the definitions of injective, surjective and bijective.

For that, let us consider a function . Each member of A is mapped to one member of B through the given function.

Injective or one-to-one means, that no two A's will be mapped to the same B. That is, many-to-one is not okay. Every B has only one A corresponding to it.

Surjective or onto means that every B has atleast one matching A. That is, every B has atleast one A mapped to it or every B has an pre-image in A. No B is left without a corresponding A.

For a function to be Bijective, it should be both Injective and Surjective, That is, it should be both one-to-one and onto at the same time.

Consider, the function . And consider such that,

So, if and only if . There the function is one-to-one or injective.

For surjective, we consider where and there should exist a x corresponding to every value of y.

If we consider y=-1, then the equation becomes :

which is not possible. So we don't get a corresponding value of x for y=-1.

Hence, the function is not surjective.

Since, the function is injective but not surjective, it is not a bijection.