# Functions

# Injective, Surjective and Bijective

A function is a way of matching the members of a set "A" to a set "B":

• A General Function points from each member of "A" to a member of "B".

  It never has one "A" pointing to more than one "B", so one-to-many is not OK in a function (so something like "f(x) = 7 or 9" is not allowed)

  But more than one "A" can point to the same "B" (many-to-one is OK)

• Injective means we won't have two or more "A"s pointing to the same "B".

  So many-to-one is NOT OK (which is OK for a general function).

  As it is also a function one-to-many is not OK

  But we can have a "B" without a matching "A"

  Injective is also called "One-to-One"

• Surjective means that every "B" has at least one matching "A" (maybe more than one).

  There won't be a "B" left out.

• Bijective means both Injective and Surjective together.

  So there is a perfect "one-to-one correspondence" between the members of the sets.

  (But don't get that confused with the term "One-to-One" used to mean injective).

# On A Graph

So let us see a few examples to understand what is going on.

When A and B are subsets of the Real Numbers we can graph the relationship.

Let us have A on the x axis and B on y, and look at our first example:

This is not a function because we have an A with many B. It is like saying f(x) = 2 or 4

It fails the "Vertical Line Test" and so is not a function. But is still a valid relationship, so don't get angry with it.

Now, a general function can be like this:

A General Function

It CAN (possibly) have a B with many A. For example sine, cosine, etc are like that. Perfectly valid functions.

But an "Injective Function" is stricter, and looks like this:

"Injective" (one-to-one)

In fact we can do a "Horizontal Line Test":

So:

• If it passes the vertical line test it is a function
• If it also passes the horizontal line test it is an injective function