2.3 Function types

The relation between two sets under a rule has two perspectives. We can look at the relation in the direction from domain set “A” to co-domain set “B”. This is the function view. But, we can also look this relation in opposite direction from “B” to “A”.

When we see function relation from domain “A” to co-domain “B”, then we find following possibilities :

• Every element of domain is related to different element of co-domain ( one to one function or injection )
• More than one elements of domain is related to an element of co-domain ( many to one function)

When we see relation from co-domain “B” to domain “A”, then we find following possibilities :

• There are elements in co-domain, which are not related to any of the elements in domain ( into function ).
• There are no elements in co-domain, which are not related to elements in domain ( onto function or surjection ).
• There are elements in co-domain, which are related to exactly one element in domain. This statement is an equivalent statement ( one to one function ).
• There are elements in co-domain, which are related to more than one element in domain. This statement is an equivalent statement ( many to one function ).

Thus, we see that there are many ways in which a function - as a relation - can be unique and hence different. This gives rise to function types, which – as we shall see – are reflection of different possibilities that we have enumerated above.

One - one function (injection)

As is evident, this function describes relation in which something can be related distinctly to something. The countries have unique and distinct capital. It is evident that a function, based on this relation, would be an injection.

One - one function (Injection)

A function $f:A\to B$ is an injection, if different elements of domain set “A” have different images in co-domain set “B”.

In plain words, every “x” in “A” associates with a distinct “y” in “B”. We can yet put it like this : Every argument (x) is related to distinct value (y).

In order to represent the condition of injectivity symbolically, we can think of two different elements “x” and “y” in set “A”. Then, two images f(x) and f(y) in “B”, corresponding to these elements in “A” are not equal. We capture this intent in constructing condition for an injection as :

$$f:A\to B\text{is an injection}\iff x\ne y,f\left(x\right)\ne f\left(y\right)\text{for all}x,y\in A$$

We can also interpret injection by asserting that if two images are equal, then it means that they are images of the same pre-image. The map diagram, corresponding to an injection, is shown in the figure. Note that elements in “A” are mapped to different elements in “B”.

Example

Problem 1: Consider a function defined as :

$$f:Z\to Zbyf\left(x\right)=x^2+1\text{for all}x\in Z$$

Determine whether the function is an injection?

Solution : We consider two arbitrary elements of the domain set such that :

$$f\left(x\right)=f\left(y\right)$$

We have deliberately considered a contradictory supposition with respect to the requirement of injectivity. If this supposition yields x = y, then the given function is an injection; otherwise not. Here,