Questions tagged [discrete-mathematics]

The study of discrete mathematical structures. Consider using a more specific tag instead, such as: (combinatorics), (graph-theory), (computer-science), (probability), (elementary-set-theory), (induction), (recurrence-relations), etc.

Discrete mathematics is not the name of a branch of mathematics, like number theory, algebra, calculus, etc. Rather, it's a description of a set of branches of math that all have in common the feature that they are "discrete" rather than "continuous".

The term "discrete mathematics" is therefore used in contrast with "continuous mathematics," which is the branch of mathematics dealing with objects that can vary smoothly (and which includes, for example, calculus). Whereas discrete objects can often be characterized by integers, continuous objects require real numbers.

Though there cannot be a definite number of branches of Discrete Mathematics, the following topics are almost always covered in any study regarding this matter −

  • Sets, Relations and Functions
  • Mathematical Logic
  • Group theory
  • Counting Theory
  • Probability
  • Mathematical Induction and Recurrence Relations
  • Graph Theory
  • Trees
  • Boolean Algebra

For an overview, see the Wikipedia entry on Discrete mathematics.


Consider using a more specific tag instead, such as: , , , , , , , , etc.

30397 questions
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How many fours are needed to represent numbers up to $N$?

The goal of the four fours puzzle is to represent each natural number using four copies of the digit $4$ and common mathematical symbols. For example, $165=\left(\sqrt{4} + \sqrt{\sqrt{{\sqrt{4^{4!}}}}}\right) \div .4$. If we remove the restriction…
6 answers

Using proof by contradiction vs proof of the contrapositive

What is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proves by contradiction, and the other proves the…
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Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$

I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals: $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$ I really have no idea why this statement is true. Can someone…
Nathan Osman
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What is the smallest unknown natural number?

There are several unknown numbers in mathematics, such as optimal constants in some inequalities. Often it is enough to some estimates for these numbers from above and below, but finding the exact values is also interesting. There are situations…
11 answers

What is the best book for studying discrete mathematics?

As a programmer, mathematics is important basic knowledge to study some topics, especially Algorithms. Many websites, and my fellows suggest me to study Discrete Mathematics before going to Algorithms, so I want to know which Discrete Mathematics…
10 answers

Would you ever stop rolling the die?

You have a six-sided die. You keep a cumulative total of your dice rolls. (E.g. if you roll a 3, then a 5, then a 2, your cumulative total is 10.) If your cumulative total is ever equal to a perfect square, then you lose, and you go home with…
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What is $\gcd(0,0)$?

What is the greatest common divisor of $0$ and $0$? On the one hand, Wolfram Alpha says that it is $0$; on the other hand, it also claims that $100$ divides $0$, so $100$ should be a greater common divisor of $0$ and $0$ than $0$.
9 answers

What is the proof that the total number of subsets of a set is $2^n$?

What is the proof that given a set of $n$ elements there are $2^n$ possible subsets (including the empty-set and the original set).
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Why is the Traveling Salesperson Problem "Difficult"?

The Traveling Salesperson Problem is originally a mathematics/computer science optimization problem in which the goal is to determine a path to take between a group of cities such that you return to the starting city after visiting each city exactly…
2 answers

A comprehensive list of binomial identities?

Is there a comprehensive resource listing binomial identities? I am more interested in combinatorial proofs of such identities, but even a list without proofs will do.
26 answers

What are some mathematically interesting computations involving matrices?

I am helping designing a course module that teaches basic python programming to applied math undergraduates. As a result, I'm looking for examples of mathematically interesting computations involving matrices. Preferably these examples would be…
11 answers

7 fishermen caught exactly 100 fish and no two had caught the same number of fish. Then there are three who have together captured at least 50 fish.

$7$ fishermen caught exactly $100$ fish and no two had caught the same number of fish. Prove that there are three fishermen who have captured together at least $50$ fish. Try: Suppose $k$th fisher caught $r_k$ fishes and that we…
20 answers

Coin flipping probability game ; 7 flips vs 8 flips

Your friend flips a coin 7 times and you flip a coin 8 times; the person who got the most tails wins. If you get an equal amount, your friend wins. There is a 50% chance of you winning the game and a 50% chance of your friend winning. How can I…
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Why are asymptotically one half of the integer compositions gap-free?

Question summary The number of gap-free compositions of $n$ can already for quite small $n$ be very well approximated by the total number of compositions of $n$ divided by $2$. This question seeks to understand why. The details A composition of an…
2 answers

An illusionist and their assistant are about to perform the following magic trick

Let $k$ be a positive integer. A spectator is given $n=k!+k−1$ balls numbered $1,2,\dotsc,n$. Unseen by the illusionist, the spectator arranges the balls into a sequence as they see fit. The assistant studies the sequence, chooses some block of $k$…
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