Exercise 3.1

If we take the monoid of the set of truth-values \(\{T, F\}\) together with \(\land\), we can write

\[F \land x = T\]

which is unsatisfiable; the equation does not hold for either of the two possible values of \(x \in \{T, F\}\).

Another example is the monoid of strings, i.e. the Monoid String instance in PureScript. The following equation is unsatisfiable, for any possible x :: String value:

"abc" <> x = "def"

One final example is the monoid \((\mathbb{N}, \max)\), where \(\max(x, y)\) is defined to be the larger of \(x\) and \(y\). Then this equation is unsatisfiable:

\[\max(5, x) = 4\]

If \(x \leq 5\), then \(\max(5, x) = 5\). If \(x > 5\), then \(\max(5, x) = x\). Either way, the result can never be \(4\).