It is currently Sun Mar 18, 2018 6:28 pm

 All times are UTC [ DST ]

 Page 1 of 1 [ 2 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Number of binary operationsPosted: Mon Jan 18, 2016 4:15 am
 Team Member

Joined: Mon Nov 09, 2015 1:36 am
Posts: 443
Location: Ioannina, Greece
How many distinct commutative binary operations can be defined in a set of $n$ elements?

NOTE: I don't have a solution.

_________________
Grigorios Kostakos

Top

 Post subject: Re: Number of binary operationsPosted: Mon Jan 18, 2016 4:16 am

Joined: Mon Nov 09, 2015 11:52 am
Posts: 76
Location: Limassol/Pyla Cyprus
Let us denote by $S =\{x_1,\ldots,x_n\}$ our set. We need to define all $x_i \ast x_j$ for each $1 \leqslant i,j \leqslant n$ and the only restrictions are that $x_i \ast x_j = x_j \ast x_i$. So $\ast$ is uniquely determined by the definitions of $x_i \ast x_j$ for each pair $(i,j)$ with $1 \leqslant i \leqslant j \leqslant n$.

Of course there are $n^{\binom{n}{2} + n} = n^{\binom{n+1}{2}.}$ such choices.

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 2 posts ]

 All times are UTC [ DST ]

Mathimatikoi Online

Users browsing this forum: No registered users and 2 guests

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forumYou cannot post attachments in this forum

Search for:
 Jump to:  Select a forum ------------------ Algebra    Linear Algebra    Algebraic Structures    Homological Algebra Analysis    Real Analysis    Complex Analysis    Calculus    Multivariate Calculus    Functional Analysis    Measure and Integration Theory Geometry    Euclidean Geometry    Analytic Geometry    Projective Geometry, Solid Geometry    Differential Geometry Topology    General Topology    Algebraic Topology Category theory Algebraic Geometry Number theory Differential Equations    ODE    PDE Probability & Statistics Combinatorics General Mathematics Foundation Competitions Archives LaTeX    LaTeX & Mathjax    LaTeX code testings Meta