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First published on Monday, Mar 9, 2026 and last modified on Monday, Mar 9, 2026
The Totally Order Set (N,≤)
And now we will arrange the integers in increasing order, to build the totally ordered set N with less or equal relationship.
Let's study the relation less or equal in N.
It is an order relationship because it has the following properties.
It is reflexive, because for any a in N, a is less or equal than a, because a is equal to a.
It is antisymmetric: If we have both a less or equal than b and b less or equal than a, then a is equal to b, because a cannot be both to the left and to the right of b unless a equals b.
It is transitive because if a is less or equal than b and b is less or equal than c, then a is less or equal than c.
This is because if a is to the left of b and b is to the left of c, then a is to the left of c.
But this is valid only for strict inequalities.
An extensive proof is in the recapitative downloadable text lecture 7.
What is the totally ordered set N with less or equal? It is an ordered set because less or equal is an order relationship in N.
It is totally ordered because we can compare any two natural integers a and b.
Either a is equal to b and then a is less or equal than b and b is less or equal than a.
Or a is to the left of b and then a is strictly less than b, so that a is less or equal than b.
Or else a is to the right of b, so that a is strictly greater than b so that b is strictly less than a, and then b is less or equal than a.
Now let's order natural integers in the strict sense.
The following properties hold for any a, b and c in N.
If a is equal to b, then neither a is strictly less than b, nor b is strictly less than a.
If they are different, then either a is strictly less than b or a is strictly greater than b, but both are never true together.
Strictly less is a transitive relation.
If a is strictly less than b and b is strictly less than c, then a is strictly less than c.
And the relation strictly greater is also transitive.
The consequences of the previous properties are the following.
You can list more than two distinct natural integers in an increasing order, for instance the digits from 0 to 9: 0, strictly less than 1, strictly less than 2, etc., up to 9.
And you can do the same in the decreasing order.
For instance, 1 million is strictly greater than 100,000, that is strictly greater than 10,000, that is strictly greater than 1,000, that is strictly greater than 100, that is strictly greater than 10, that is strictly greater than 1.
Let's see some more properties of the natural integer strict ordering.
And let's consider a and b natural integers.
If a is not zero, then a is strictly greater than zero, or strictly positive.
That's why N*, natural integers that are non-zero, are called the positive integers.
If b is the follower of a, then b is strictly greater than a, and there is no other natural integers in between.
The elements of N follow each other in an increasing order.
And there is no negative natural integers, the smaller natural integer is zero.
Strict and non-strict comparison operators share mutual properties.
For a and b natural integers, the following equivalences hold.
a is less or equal than b if and only if a is not greater than b.
And a is greater or equal than b if and only if a is not less than b.
The consequence of that is that, as there is no natural integers that is strictly less than zero, all natural integers are positive of zero, they are non-negative.
Do we go up to infinity? We know that any natural integer n has a follower n+, and that follower is a natural integer.
n+ is in N and it is strictly greater than n n+ is strictly greater than n.
So we go up to infinity, because the set N is an everlasting increasing sequence.
It has no maximum, because a maximum m would have a follower m+, that would be greater strictly than m.
That is in contradiction with the fact that m is a maximum.
So yes, N starts from zero and goes to plus infinity.
And we shall end this lecture with the sequences of integers.
A sequence of natural integers is an infinite list.
u equals u_n, with n positive or zero, with n non-negative, natural integer.
It is indexed by the natural integers n with elements u_n in N, natural integers also.
The examples of sequences of natural integers are the following.
u_n equals n.
v_n equals 2 times n.
And w_n is n to the square.
But let's define and draw sequences in Python.
Launch Anaconda and Spyder and follow me.
For that we shall use NumPy numerical computation package: from numpy import star.
So that we import all the functions of numpy package, library.
from numpy import *And then we define a vector u made of zeros, indexed from 0 to 99, with zeros.
And then we shall fill u with the sequence u_n equals n.
For n in arange of 100, the index is from 0 to 99, u of n, into brackets, is equal to n.
So we define the sequence u_n = n.
u=zeros([100])
from n in arange(100):
u[n]=nSo we define the sequence u_n = n.
Now we shall plot it. But before that we have to import the package matplotlib.pyplot, in order to have the function plot.
import *.
from matplotlib.pyplot import *And then we plot u.
That's it! And we have a straight line from 0 to 100, in x and in y.
plot(u)Now we define the vector v with zeros and we fill v with v of n into brackets equals 2 times, 2 star n, 2 times n.
v_n is equal to 2 times n.
v=zeros([100])
from n in arange(100):
v[n]=2*nNow we plot v, and we have a straight line that goes up to two hundredths in the ordinate.
plot(v)Now we define the vector w with zeros, and we fill it w of n into brackets is equal to n to the power, star star, to n to the square.
v=zeros([100])
from n in arange(100):
v[n]=n**2And now we plot w, and we have an arch of parabola.
It goes to 10,000 in the ordinate.
plot(w)What's coming now? What's coming now is a recapitulative document with extensive proofs and complements.