Lecture 18 : One-to-One and Onto Functions.
In this lecture, we will consider properties of functions:
Functions that are One-to-One, Onto and Correspondences.
Comparing cardinalities of sets using functions.
One-to-One/Onto Functions
Here are the definitions:
Example-1
Classify the following functions between natural numbers as one-to-one and onto.
| One-to-One? | Onto? |
| Yes | No |
| Yes | No |
| No | Yes |
. | Yes | Yes
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It helps to visualize the mapping for each function to understand the answers.
Example-2
Prove that the function is one-to-one.
Proof: We wish to prove that whenever then . Let us assume that for two numbers . Therefore,
. Which means that . Splitting cases on , we have
Example-3
Prove that the function is onto.
Claim-1 The composition of any two one-to-one functions is itself one-to-one.
Claim-2 The composition of any two onto functions is itself onto.
Comparing Cardinalities of Sets
Let and be two finite sets such that there is a function . We claim the following theorems:
The observations above are all simply pigeon-hole principle in disguise.
Theorem Let be two finite sets so that . Any function from to cannot be one-to-one.
Proof
Let be any function. Think of the elements of as the holes and elements of
as the pigeons. There are more pigeons than holes. Therefore two pigeons have to share (here map on to) the same hole.
QED.
We now prove the following claim over finite sets .
Claim Let be a finite set. Prove that every one-to-one function is also onto.
Proof
We will prove by contradiction.
The last statement directly contradicts our assumption that is one-to-one.
QED.
Infinite Sets
We now note that the claim above breaks down for infinite sets.
Let us take , the set of all natural numbers.
Hilbert's Hotel
Consider a hotel with infinitely many rooms and all rooms are full.
An important guest arrives at the hotel and needs a place to stay. How does the manager accommodate the new guests even if all rooms are full?
Each one of the infinitely many guests invites his/her friend to come and stay, leading to infinitely many more guests. How does the manager accommodate these infinitely many guests?
One-to-One Correspondences of Infinite Set
There is a one to one correspondence between the set of all natural numbers and the set of all odd numbers .
Take , where .
We note that is a one-to-one function and is onto.
Can we say that ? Yes, in a sense they are both infinite!! So we can say !!
Therefore we conclude that
There are “as many” even numbers as there are odd numbers?
There are “as many” positive integers as there are integers? (How can a set have the same cardinality as a subset of itself? :-)
There are “as many” prime numbers as there are natural numbers?
Note that “as many” is in quotes since these sets are infinite sets.
Infinite Sets
There are many ways to talk about infinite sets. We will use the following “definition”:
A set is infinite if and only if there is a proper subset and a one-to-one onto (correspondence) .
Here are some examples of infinite sets:
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