Question

For which operations is the set {0, 1} closed?

Answer 1

Are you restricted to just addition and multiplication? The answer would depend on the binary operators you're given. We can define our own operation such that the given set is closed under said operation.

For example, I can define an operation \(\%:S\to S\) such that for any \(a,b\in S\) you have \(a\%b=a\). So if \(S=\{0,1\}\), and \(a=0\) and \(b=1\), then
\[\begin{array}{c|c|c}
&a&b\\
\hline
a&a&a\\
\hline
b&b&b
\end{array}\]
(where the entry in the left column is written before \(\%\).)