Cameron S.

Mathematics major 🖖

[leibnizstan]

“You can either believe in the axiom of choice and live a happy life or choose not to believe in it and die miserable”

-math professor

[cams-math-rants]

Intro to Analysis

I've started studying real analysis over the summer since it is a class I will be taking next semester, and I am greatly enjoying it so far. I would like to share an interesting proof I read about.

The proof I am showing is that ℚ is dense in ℝ:

(a) x,y∈ℝ and x<y ⇒ ∃p∈ℚ such that x<p<y

Which means for every two real numbers x,y where x<y, you can always find a rational number between them.

To understand this proof, you need to know about the Archimedean property:

(b) x,y∈ℝ and x>0 ⇒ ∃n∈ℤ+ such that nx>y

-----------------------

PROOF:

Since x<y, this implies that 0<y-x. By (b), we see that ∃n∈ℤ+ such that (1/n)<y-x, or that 1<n(y-x)=ny-nx.

When applying (b) again, we have that ∃m1,m2∈ℤ+ such that m1>nx and m2>-nx. Then, we get -m2<nx<m1.

We know that ∃m∈ℤ, where m2≤m≤m1, such that m-1≤nx<m.

Then, nx<m≤nx+1.

Since 1<ny-nx implies ny>nx+1, we have that nx<m≤nx+1<ny.

Since n>0, it follows that x<(m/n)<y.

This proves (a) for p=(m/n).

∎

What would you like to see next

Proof of the Archimedean property

Understanding the density of ℚ in ℝ (involving √2)

Voting ended onMay 15, 2025

Sources:

Rudin, Walter. Principles of Mathematical Analysis. 1953. 3rd ed., ‎McGraw-Hill Publishing Company.

The song I'm currently listening to:

[cams-math-rants]

Finite and Infinite Sets

Recently, in my Discrete Mathematics course, we have been discussing the title of this blog post. It took me some time to understand these concepts, but it is finally clicking in my mind and it’s all I can think about at the moment.

First, I’d like to discuss some definitions.

When I mention the notion of a set, this refers to a collection of objects/numbers (we call each number in a set an “element”) in which the order doesn’t matter (unlike with sequences). An example of a set is ℝ, the set of all real numbers. Another example that will become important later is ℕ, the set of natural numbers (positive integers starting at 1).

A bijection refers to when every element in one set maps onto a unique element in the other set. When we map a set onto another, we write X → Y. This is shown in the image below, because it helps to see this visually.

When two sets are equinumerous (written X ≈ Y for two sets X and Y), this simply means there is a bijection between these sets.

By definition, in order for a bijection to be possible, the two sets have to contain the same amount of elements, or have the same cardinality (written |X| = |Y|, which in English means “the cardinality of X equals the cardinality of Y”). This means the two sets have the same size.

Now, we can talk about finite and infinite sets. The notion of a finite set is rather intuitive: there are n elements in a set X where |X| = n. An infinite set also makes sense conceptually, where the elements never reach a boundary. However, there are different kinds of finite and infinite sets that makes things interesting.

We can say a set is denumerable, which means we get a bijection when mapping ℕ to said set. An example of this is the set of integers, ℤ. When we have ℕ → ℤ. We can show this by representing this as a piecewise function and separating the integers into even and odd, and we find that there is a one-to-one, unique correspondence between the two sets (perhaps I will go through this proof in another post but there are examples online).

A denumerable set can be infinite or finite. In the example above, ℤ is infinite and denumerable. When a set is countable, that means a set is either finite or denumerable.

What is interesting is when we have an infinite and uncountable set, such as ℝ. In this case, there is a beautiful way to prove such a thing using Cantor’s Diagonal Argument. I won’t discuss this in full here, but here is the link to the Wikipedia article: https://en.wikipedia.org/wiki/Cantor's_diagonal_argument?wprov=sfti1

Cantor's diagonal argument - Wikipedia

Cantor's diagonal argument - Wikipedia

Thanks to anyone who has read my post. My name is Cameron and I’m a math major that loves to discuss what I’m learning in my classes. Keep in mind that I am not an expert, so I am open to anyone scrutinizing my explanations if you believe something is represented wrongly.

Would anyone want more posts from me about various math subjects?

Yes

No

Voting ended onApr 17, 2025

This is the song I’m currently listening to on repeat:

[ettolrach-uwu]

I don't think it's true that a "denumerable" set can be finite? I've not heard of the term before, but using your definition, a set S is denumerable iff there exists a bijection f: S → ℕ. But that immediately implies that S must be infinite, as a bijection implies that S and ℕ are isomorphic.

By the way, if you enjoy set theory then you should definitely learn of the Cantor-Bernstein theorem (also known by many different names of mathematicians) which says that, given two sets A, B, if there exists an injection f: A → B and an injection g: B → A, then A is isomorphic to B. This often makes proofs much easier, as you don't need to find a bijective function, just two injective ones going each way. So the proof of ℕ ≅ ℤ is: let f: ℕ → ℤ be the inclusion function, which is clearly injective. Note how ℤ ≅ ℕ × ℕ because we can interpret members of ℤ as the difference of two naturals. We now just need to show that there exists an injective g: ℕ × ℕ → ℕ. We define it as g(a, b) = 2^a 3^b. Then by the uniqueness of prime factorisation, g is injective, QED.

[cams-math-rants]

You’re right. I’m still learning these topics and I believe I mixed up some definitions. I meant to say a countable set can be finite or infinite if it is denumerable. Thanks for letting me know and I will look into the Cantor-Bernstein theorem.

[cams-math-rants]

Please tell me how you fell in love with math

I would love to answer this!

I didn’t always love mathematics, in fact most of my life I had a good amount of disdain for those classes. To me, it seemed somewhat pointless, or at least I couldn’t see how it would be useful. Then, my junior year of high school, I became fascinated with physics. I began reading up on so many astrophysics books and it was all I could think about. This is when I started to realize how useful and even fun math could be.

I entered university as a physics major and philosophy minor initially. As I took all the required math courses I needed for physics, I became more and more interested. I changed my minor to math, and then my sophomore year of college is when I officially switched my major. I was comparing all the physics classes I would take to the math classes I could take if it was my major and I was so much more compelled by the latter.

Ever since I’ve been in love with my studies. My dream is to be a high school mathematics instructor.

[cams-math-rants]

An Introduction to Number Theory

The last poll I posted was to choose between set theory and number theory, and the latter one received the most votes, so that will be the basis for this post.

Number theory is the branch of pure mathematics concerned with the study of integers and their properties, and it is one of the oldest fields of research.

I will talk about the first couple of notable concepts I learned in my classes: The Division Algorithm, Greatest Common Divisors, and Euclidean’s Algorithm.

The Division Algorithm (also called the Division-Quotient Theorem)

This is basically a formal way of defining long division, but it is useful in certain ways, as we will get to one of these.

Ex.) Let’s say we are given two integers a=8 and b=3. What unique integers q and r are there (q being the quotient and r being the remainder)?

8 = 3q + r

3 goes into 8 twice with 2 left over, so q,r=2

8 = 3(2) + 2

The Greatest Common Divisor (GCD)

*Note on notation: x | y means that x divides y

(Like 3 | 6 because 3 goes into 6 twice)

The Euclidean Algorithm

This is a way to find the GCD of two positive integers. I will represent the informal definition here because, formally, it is a bit notationally complicated.

Take note of part (3), where it says (a,b) = (b,r). This comes from a lemma that follows from the Division Algorithm that is useful for the EA. I have provided this lemma below:

Ex.) Find gcd(58,17)

(1) We are given two positive integers 58 and 17

(2) 17 does not divide 58, so we can move on.

(3) Use the Division Algorithm:

58 = 17q + r → 58 = 17(3) + (7) → (58,17) = (17,7)

17 = 7q + r → 17 = 7(2) + 3 → (17,7) = (7,3)

7 = 3q + r → 7 = 3(2) + 1 → (7,3) = (3,2)

3 = 2q + r → 3 = 2(1) + 1 → (3,2) = (2,1)

2 = q + r → 2 = (1) + 1 → (2,1) = (1,1)

1 = q + r → 1 = (1) + 0 → (1,1) = (1,0) = 1

So 1 is the greatest common divisor of 58 and 17. This is a special case. When we find that the GCD of two positive integers is 1, we can call them Coprime (also called Relatively Prime).

That’s all I have to say for now and I hope you find something interesting in this post. If you have any questions I will be glad to (attempt to) answer them.

Resources:

The Tools of Mathematical Reasoning by Tamara J. Latkins

The song I’m listening to at the moment:

[cams-math-rants]

Choose one

Set Theory

Number Theory

Voting ended onApr 18, 2025

[cams-math-rants]

Finite and Infinite Sets

Recently, in my Discrete Mathematics course, we have been discussing the title of this blog post. It took me some time to understand these concepts, but it is finally clicking in my mind and it’s all I can think about at the moment.

First, I’d like to discuss some definitions.

When I mention the notion of a set, this refers to a collection of objects/numbers (we call each number in a set an “element”) in which the order doesn’t matter (unlike with sequences). An example of a set is ℝ, the set of all real numbers. Another example that will become important later is ℕ, the set of natural numbers (positive integers starting at 1).

A bijection refers to when every element in one set maps onto a unique element in the other set. When we map a set onto another, we write X → Y. This is shown in the image below, because it helps to see this visually.

When two sets are equinumerous (written X ≈ Y for two sets X and Y), this simply means there is a bijection between these sets.

By definition, in order for a bijection to be possible, the two sets have to contain the same amount of elements, or have the same cardinality (written |X| = |Y|, which in English means “the cardinality of X equals the cardinality of Y”). This means the two sets have the same size.

Now, we can talk about finite and infinite sets. The notion of a finite set is rather intuitive: there are n elements in a set X where |X| = n. An infinite set also makes sense conceptually, where the elements never reach a boundary. However, there are different kinds of finite and infinite sets that makes things interesting.

We can say a set is denumerable, which means we get a bijection when mapping ℕ to said set. An example of this is the set of integers, ℤ. When we have ℕ → ℤ. We can show this by representing this as a piecewise function and separating the integers into even and odd, and we find that there is a one-to-one, unique correspondence between the two sets (perhaps I will go through this proof in another post but there are examples online).

A denumerable set can be infinite or finite. In the example above, ℤ is infinite and denumerable. When a set is countable, that means a set is either finite or denumerable.

What is interesting is when we have an infinite and uncountable set, such as ℝ. In this case, there is a beautiful way to prove such a thing using Cantor’s Diagonal Argument. I won’t discuss this in full here, but here is the link to the Wikipedia article: https://en.wikipedia.org/wiki/Cantor's_diagonal_argument?wprov=sfti1

Cantor's diagonal argument - Wikipedia

Cantor's diagonal argument - Wikipedia

Thanks to anyone who has read my post. My name is Cameron and I’m a math major that loves to discuss what I’m learning in my classes. Keep in mind that I am not an expert, so I am open to anyone scrutinizing my explanations if you believe something is represented wrongly.

Would anyone want more posts from me about various math subjects?

Yes

No

Voting ended onApr 17, 2025

This is the song I’m currently listening to on repeat: