Doubt about category theory exercise on Bool monoid

Question

I'm studying Category Theory for Programmers.

At the end of chapter 3, the exercise number 3 asks

Considering that Bool is a set of two values True and False, show that it forms two (set-theoretical) monoids with respect to, respectively, operator && (AND) and || (OR).

but I'm not sure about what to do.

For sure I can see the parallel between the monoids $(\Bbb N, +, 0)$ and $(\Bbb N, \times, 1)$, and the (candidate) monoids $(\Bbb B, \lor, F)$ and $(\Bbb B, \land, T)$, where I'm using $\Bbb B = \{T, F\}$, because it's easy to verify that properties of a monoid hold. (If the symbols, which I have seen somewhere, are not right, please feel free to correct me.)

But is it just this? I mean, is solving the exercise just recognizing this in the paragraph above?

Answer 1

Yes, they are somewhat similar.

You can directly prove the associativity of the given operations and that the given element is their identity.

Or, you can work with these isomorphic analogies: consider the monoids $$(\{0,1\},\min)\ \text{ and }\ (\{0,1\},\max)$$

Note also that these two operations satisfy even more properties: they are commutative, idempotent ($a\lor a=a=a\land a$) and also distributive with respect to each other.