Why Discrete Mathematics is a Challenging College Course

The World’s Best Mathematician (*) – Numberphile
The World’s Best Mathematician (*) – Numberphile

There is no way around discrete maths if you plan to major in computer science or applied mathematics. But why do so many students fear discrete mathematics?

When I was studying mathematics, fellow students found discrete maths challenging because it revolves around discrete (non-continuous) objects, which differs from most “traditional” courses like statistics and calculus. This new paradigm means students are “starting from scratch” and cannot relate to prior knowledge.

At the highest level of college mathematics, discrete mathematics is even more challenging and requires students to understand various techniques from combinatorics, probability, and graph theory.

Discrete Mathematics Concepts Are Hard to Grasp for Fresh High School Graduates

For the most part, high school students are taught to either think algebraically or use the continuous framework taught in Calculus. This is why their a bit disoriented when they are asked to master new concepts.

This does not mean that discrete mathematics is more difficult, relatively speaking, but it does require a different train of thought to wrap your head around it.

Discrete Vs. Continuous Mathematics

Discrete mathematics deals with… discrete objects.

To help you understand the concept of discreteness, I like to give the following non-formal example:

  • Natural numbers (1, 2, 3, 4, 5, etc., up to infinity) are “discrete.” There is no other number between the numbers 1 and 2. We could say there is a sharp transition between these numbers because there is no continuous change.
  • Real numbers are “continuous.” In the real set, if you take two real numbers, like 1 and 2, you can always find an infinity or other real numbers between them. There is no “sharp” transition between any two elements of the set.

Think of discrete maths as a digital clock with numbers transitioning to the next abruptly, while an analog clock (continuous) would have the hands move almost smoothly over time.

In My Experience, These Are The 3 Most Challenging Discrete Mathematics Topics

Discrete mathematics is a vast field. Some important topics you need that are discussed in discrete maths are Graph Theory, Combinatorics, Probability, Algorithms, Boolean Algebra, and Matrices.

Below are the three courses I found the most challenging during my first discrete mathematics courses.

1. Graph Theory

A graph only consists of two elements: vertices and edges. Vertices are the individual objects in the graph, while edges are the connections or relationships between them. For example, the internet is a graph where each router is a vertex, and each internet cable between routers is an edge.

In graph theory, there are plenty of problems that can be asked with only the most basic of definitions but are incredibly difficult to solve, such as:

  • Given a graph, can it be colored with only two colors such that no two adjacent vertices have the same color? This is known as the “two-coloring problem” or the “bipartite graph problem.”
  • Given a graph, can it be traversed starting at a given vertex and visiting every other vertex exactly once? This is known as the “Eulerian path problem.”
  • Given a graph, can it be traversed starting at a given vertex and visiting every other vertex exactly once, and then returning to the starting vertex? This is known as the “Eulerian circuit problem.”

You might be able to solve them if you are a math wizard (or if you read the solution out of a textbook).

2. Algorithms and Optimization Problems

When I was in college, we studied discrete math algorithms to solve optimization problems. This requires advanced mathematical optimization techniques, such as linear programming, convex optimization, and dynamic programming.

Algorithms in discrete mathematics can be used to optimize supply routes, reorganize cellphone network towers, and other optimization problems.

3. Combinatorics

Combinatorics deals with abstract concepts such as permutations, combinations, and partitions, which can be difficult to understand for students unfamiliar with advanced mathematics.

There are many different types of combinatorial problems, each requiring a different approach and set of techniques. This can make it challenging for students to understand which technique to use for a given problem.

The Bottom Line

Discrete maths mainly focuses on non-continuous objects in relation to maths, while this is the opposite of calculus and algebra. This is why some students may find it challenging to grasp at first because they have to think differently to problem-solve.

Although faced with this issue, discrete math is logical. Unlike other math courses, grasping the fundamentals for many students sometimes seems easier than calculus and algebra. Instead of memorizing and working through practice examples, puzzle-solving (with logic) will get you through discrete maths.

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