Consider the operation $*$ on the set of reals $\mathbb{R}$ defined as follows:
$a*b=c$, where $c$ is the smallest real number greater than both $a$ and $b$.
My question: Is the operation $*$ binary on $\mathbb{R}$?
My confusion: For an operation $*$ to be binary on set $S$, the closure property must be satisfied. I know that the result $a*b=c$ surely belongs to $\mathbb{R}$ but can we say that the operation is well-defined for any pair $(a,b)$ as giving such $c\in\mathbb{R}$ explicitly is not possible? Thanks.
A binary operation on a set $S$ is simply a function $f : S\times S\to S$. You need to check that $(a,b)\mapsto a\ast b$ is well-defined (i.e., actually a function whose codomain is $\Bbb R$).
The ``smallest real number greater than a given real number" is a problematic concept: there is no smallest real number greater than a given number! For example, consider $0$. What is the smallest real number greater than $0$? It would have to be the smallest positive number. But suppose there was one, and call it $\epsilon$. Then $0 < \epsilon/2 < \epsilon$, so $\epsilon/2$ is a smaller positive number than $\epsilon$, contradicting our assumption that $\epsilon$ was the smallest! A small modification of your argument shows that $a\ast b$ is not a well-defined real number (that is, no real number exists satisfying the defining property of $a\ast b$), so that $\ast$ is not a binary operation on $\Bbb R$.