A binary operation defined on a set with one element

solution-verification abstract-algebra

A binary operation $\star$ defined on the set $S$ is a function $S\times S\mapsto S$, so it is closed over $S$ by definition. The idea of closure only makes sense when talking about proper subsets of $S$.

The answer to the question is yes. Suppose $\star$ is a binary operation on $\{x\}$. Then if $a,b,c\in\{x\}$ we have $ab=ba$ and $a(bc)=(ab)c$, since everything is $x$.

Related

Find marginal distribution of Y while knowing distribution of X and $Y|X$
Find kernel of induced map after tensoring
Radius of convergence of the complex serie $(z-i)^n/n!$
The $n$-th derivative of $x^2(x+1)^n$?
Is $(1-x)^{\alpha} \log(1-x)$ a Sobolev function?
A 1/x function that intesects both the x and y axes at specific points, and whose shape can be changed.
Find eigenvalue of $B= I + A^{-1}+A^{-2}+A^{-3}+...$
Bayes theorem problem example
Matrix Spectral Radius and Induced Matrix Norms
Prove that the angles between tangents to circles centered on a trapezium are equal
If $U$ commutes with $H$, can we say if U also commutes with $H^\dagger$?
Finding the first coefficients of a power series.