Injections, surjections and bijections

A function f: A \rightarrow B is called an injection if f(a)=f(a^{'}) implies a=a^{'}. There is another name for this kind of function. It is also called a one-to-one function or an injective map. A map f:A \rightarrow B is one-one if two distinct elements of A have distinct images under f.

A function f: A \rightarrow B is called a surjection if for every b \in B there is an element a \in A such that f(a)=b. In other words, f(A)=B. 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 f:A \rightarrow B which is both one-to-one and onto is called a bijection or a bijective map.


1) Suppose f: \Re \rightarrow \Re is defined by f(x) =\cos{x}. It is clear that this is neither one-to-one nor onto. Indeed because f(x)=f(x+2\pi), it cannot be one-to-one. Since f(x) never takes a value below -1 or above 1, it cannot be onto.

2) f: \Re \rightarrow \Re defined by

f(x)= x, if x \geq 0 and f(x)=x-1, if x<0 is one-to-one but not onto as -1/2 is never attained by the function.

3) f:\Re \rightarrow \Re defined by f(x)=\frac{x}{1+|x|} is one-to-one but not onto.


Trigonometric functions like sine and cosine are neither one-to-one nor onto. So, how does one define \sin^{-1}{x} or \cos^{-1}{x}? Actually, there is ambiguity in defining these. If we write \sin^{-1}{x}=\theta, it means that \sin{\theta}=x. It is easily seen that there is no \theta if x is more than 1 or less than -1. Thus, the domain of \sin^{-1}{x} or \cos^{-1}{x} must be [-1.1]. Then, again

\sin{\theta}=x has many solutions \theta for the same x. For example, \sin{\pi/6}=\sin{5\pi/6}=1/2. So, which of \pi/6 or 5\pi/6 should claim to be the value of \sin^{-1}{(1/2)}? In such a case, we agree to take only one value in a definite way. For 0 \leq x \leq 1, we choose 0 \leq \theta \leq \pi/2 such that \sin{\theta}=x. It is obvious that there is only one such \theta. Similarly, for -1 \leq x <0, we choose -\pi/2 \leq \theta < 0 such that \sin{\theta}=x. Thus, this way of choosing \theta such that \sin{\theta}=x for -1 \leq x <1 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 f:\Re \rightarrow \Re defined by f(x)=x^{2}. The function f is neither one-to-one nor onto. But, if we take f: \Re \rightarrow \Re_{+} as f(x)=x^{2}, then f is onto. We would like to define f^{-1}:\Re_{+} \rightarrow \Re as a function. In order that we are able to define f^{-1} as a function we must agree, once and for all, the sign of f^{-1}{(x)}. Indeed, since f(-1)=f(1)=1, which one would we call f^{-1} ? In fact, f^{-1}{(x)} is what we would like to denote by \sqrt{x}. But, we must decide if we are taking the positive value or the negative value. Once, we decide that, f^{-1} would become a function.

More later,

Nalin Pithwa




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