Discrete Math Overview

What even is discrete math? #

According to Wikipedia, “Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.” Very helpful, thank you Wikipedia. The floor is indeed made of floor rather than sky.

The word discrete means “distinct” or “countable”. This suggests that discrete math has to do with countable numbers like integers, rather than the continuous $f(x)$functions we’re used to seeing that are defined for any real$x$, even ones we don’t know the exact value of like $\pi$.

Dealing with countable integers is nice because that’s how computers work. Behind the scenes, every floating point number is actually just a whole bunch of bits, which are countable :) I would say that dealing with integers makes things nicer too (since we no longer have to deal with decimals), but you might be inclined to disagree.

A brief summary of the contents covered #

Discrete math is an extremely wide field of mathematics. Here, we’ll be covering the basics as well as a few important applications:

• Propositional logic

  What are Propositions? Propositions are anything that can be true or false. This could include: Statements like "Birds can fly". Well defined equations...

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  and sets give us the language we need to talk about discrete math.
• **** Proofs

  Introduction A proof is a set of logical deductions that can be used to show how something is true. This is...

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  **** allow us to demonstrate how and why things work the way they do.
• Stable Matching

  Introduction The stable matching problem deals with how to match one group to another group while trying to maximize everyone's 'happiness'....

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  explores how we can apply sets to create optimal matches between two groups with preferences.
• Graph theory provides a highly visual representation of a wide variety of mathematical relationships using vertices, edges, and faces. One of the most important concepts here is Euler’s Formula which relates the number of vertices, edges, and faces together.
• **** Modular arithmetic

  Modular Arithmetic

  What is Modular Arithmetic? Modular arithmetic is "clock math" - that is, when numbers wrap around back to 0 if they...

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  explores what happens when when all numbers are remainders of dividing itself by another number. There are some really important theorems here, like the Chinese Remainder Theorem, Euclid’s Algorithm, and Fermat’s Little Theorem.
• **** RSA Cryptography

  RSA Cryptography

  Introduction The internet is built upon the fact that stuff needs to go from point A to point B quickly, accurately,...

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  **** is an interesting application of how modular arithmetic is used to encrypt and decrypt messages using a public-private key pair.
• Polynomials

  Polynomials

  Introduction "I learned this in 4th grade", you say, "and I already know how to do Taylor approximations and binomial expansions...

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  can be used in a discrete sense to create secret sharing schemes, and can be recovered from points using Lagrange Interpolation.