A function is called an injection if implies . There is another name for this kind of function. It is also called a one-to-one function or an injective map. A map is one-one if two distinct elements of A have distinct images under f.

A function is called a surjection if for every there is an element such that . In other words, . That is to say, every element of B is an image of some element of A under f. A surjective map is also called an onto map.

A map which is both one-to-one and onto is called a bijection or a bijective map.

**Examples.**

1) Suppose is defined by . It is clear that this is neither one-to-one nor onto. Indeed because , it cannot be one-to-one. Since never takes a value below -1 or above 1, it cannot be onto.

2) defined by

, if and , if is one-to-one but not onto as is never attained by the function.

3) defined by is one-to-one but not onto.

**Warning.**

Trigonometric functions like sine and cosine are neither one-to-one nor onto. So, how does one define or ? Actually, there is ambiguity in defining these. If we write , it means that . It is easily seen that there is no if x is more than 1 or less than -1. Thus, the domain of or must be . Then, again

has many solutions for the same x. For example, . So, which of or should claim to be the value of ? In such a case, we agree to take only one value in a definite way. For , we choose such that . It is obvious that there is only one such . Similarly, for , we choose such that . Thus, this way of choosing such that for has no ambiguity. such a value of the inverse circular function is called its principal value, though we could choose another set of values with equal ease.

Similar problems arise in the context of a function defined by . The function f is neither one-to-one nor onto. But, if we take as , then f is onto. We would like to define as a function. In order that we are able to define as a function we must agree, once and for all, the sign of . Indeed, since , which one would we call ? In fact, is what we would like to denote by . But, we must decide if we are taking the positive value or the negative value. Once, we decide that, would become a function.

More later,

Nalin Pithwa

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