Lecture 18 : OnetoOne and Onto Functions.In this lecture, we will consider properties of functions:
OnetoOne/Onto FunctionsHere are the definitions:
Example1Classify the following functions between natural numbers as onetoone and onto.
It helps to visualize the mapping for each function to understand the answers. Reasons
Example2Prove that the function is onetoone. Proof: We wish to prove that whenever then . Let us assume that for two numbers . Therefore, . Which means that . Splitting cases on , we have
Example3Prove that the function is onto. Proof
Given any , we observe that is such that . Therefore, all are mapped onto. Claim1 The composition of any two onetoone functions is itself onetoone. Proof
Let and be both onetoone. We wish to tshow that is also onetoone.
The reasoning above shows that is onetoone. Claim2 The composition of any two onto functions is itself onto. Proof
Let and be onto functions. We will prove that is also onto.
Comparing Cardinalities of SetsLet and be two finite sets such that there is a function . We claim the following theorems:
The observations above are all simply pigeonhole principle in disguise. Theorem Let be two finite sets so that . Any function from to cannot be onetoone. 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 onetoone function is also onto. Proof
We will prove by contradiction.
The last statement directly contradicts our assumption that is onetoone. QED. Infinite SetsWe now note that the claim above breaks down for infinite sets. Let us take , the set of all natural numbers.
Hilbert's HotelConsider a hotel with infinitely many rooms and all rooms are full.
OnetoOne Correspondences of Infinite SetThere 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 onetoone function and is onto. Can we say that ? Yes, in a sense they are both infinite!! So we can say !! Therefore we conclude that
Note that “as many” is in quotes since these sets are infinite sets. Infinite SetsThere 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 onetoone onto (correspondence) . Here are some examples of infinite sets:
