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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.

and http://www.cs.yale.edu/homes/aspnes/classes/202/notes.pdf

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

32903 questions
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An interesting problem with "decomposing" natural numbers.

Consider the following one-person game: A player starts with score $0$ and writes the number $20$ on an empty whiteboard. At each step, she may erase any one integer (call it $a$) and writes two positive integers (call them $b$ and $c$) such that…
Lucius
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3 answers

An impossible sequence of Tetris pieces.

Is there a finite sequence of pieces of Tetris such that for every way of play you always lose?
user195451
19
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9 answers

Proof: If n is a perfect square, $\,n+2\,$ is NOT a perfect square

"Prove that if n is a perfect square, $\,n+2\,$ is NOT a perfect square." I'm having trouble picking a method to prove this. Would contraposition be a good option (or even work for that matter)? If not, how about contradiction?
Greg
  • 191
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3 answers

Identity relation vs Reflexive Relation

So we're starting relations in my discrete structures class this week, and I've probably read this over 10 times by now...I believe I have a good understanding of Identity Relations, but Reflexive Relations seem to have me slightly confused. From my…
Rickybobby
  • 367
18
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1 answer

Find all $A\subseteq\mathbb{N}$ such that $A=\{|a-b|:a,b\in A\}$.

For a set $A$ of real numbers, denote $$A^\ast:=\{|a-b|:a,b\in A\}.$$ Question: Find all finite subsets $A\subseteq\mathbb{N}$ of the natural numbers such that $$A^*=A.$$ Attempt: The empty set is a trivial solution, so let's assume…
Simon Parker
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8 answers

Proof: For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$

I need help proving the following statement: For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$ The statement is true, I just need to know the thought process, or a lead in the right direction. I think I might have to use a contradiction,…
Brad
  • 307
15
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11 answers

What does it mean for a number to be in a set?

Frustratingly my book gives me several examples of a number in a set but offers no explanation at all. Anyways what is going on here? According to the book $2$ is not an element of these…
Paul the Pirate
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15
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1 answer

Showing $(x,y)$ pairs exist for $\sqrt{\quad\mathstrut}$

If we were to show that there exists infinitely many $(x,y)$ pairs in $\mathbb{Q}^2$ for which both $\sqrt{x^2+y^4}$ and $\sqrt{x^4+y^2}$ are rational. If the power root for $x$ and $y$ vary but never the same, how can we prove that we can always…
user65422
  • 1,123
12
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2 answers

Can we find the GCD of a positive and negative number?

I'm trying to evaluate whether or not $\gcd(p,q) = \gcd(-p,q)$ for non-zero integers $p$ and $q$. I was wondering if it's possible to find $\gcd(-p,q)$. If so, this statement should be true, correct?
Pandamonium
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11
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Prove that there's no fractions that can't be written in lowest term with Well Ordering Principle

This is from Class Note from 6.042 ocw courses at MIT: "Well Ordering Principle" section: ( Sorry for not posting latex; I have less than 10 reputations to post images ) You can read the original here at page 1 and 2; Well Ordering…
Hieu Nguyen
  • 213
11
votes
3 answers

How do I know when to use nCr button and when to use nPr button on my calculator in which situation?

How do I know when to use nCr button and when to use nPr button on my calculator? nCr= combinations I believe NPR= permutations Is there a general rule I can use to figure out which one to use and when according to the given question?
MethodManX
  • 1,127
11
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4 answers

Proof by contradiction: $r - \frac{1}{r} =5\Longrightarrow r$ is irrational?

Prove that any positive real number $r$ satisfying: $r - \frac{1}{r} = 5$ must be irrational. Using the contradiction that the equation must be rational, we set $r= a/b$, where a,b are positive integers and…
Mr_CryptoPrime
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10
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1 answer

Expressing Powers in Terms of Falling Powers

The falling power $n^\underline{k}$ (read $n$ to the falling $k$) is defined as follows: $$n^\underline{k}=n(n-1)(n-2)\cdots(n-k+1)$$ These are important in discrete calculus because their finite differences and sums are analogous to those of normal…
David Zhang
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10
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1 answer

Arrangement of Numbers

How can we prove that it is posible to arrange numbers $1,2,3,4,\ldots, n$ in a row so that the average of any two of these numbers never appears between them?
aaaaaaaaaaaaaaaaaaaaaaaaaa
  • 141
10
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2 answers

Basic Equivalence Class Discrete Math

I read through the textbook definition of the equivalence class, but still cannot clearly understand what an equivalence class is. Does anyone have a good example with a definition that can hit me home?
Aaron
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