# Exercise 2.36#

Letâ€™s build a list of example statements (including those that were already made) that may be made in this language. This is part of specifying â€śinterpretationâ€ť as defined in forall x: 30.5 Interpretations:

\(Q\): n is prime

\(R\): n is even

\(S\): \(0 \leq n\)

\(T\): \(n = 15\)

Clearly \(S \leq P\) for all \(P\) in \(n \in â„•\), but \(T\) is only true for a single \(P\). If we use â€śandâ€ť as our monoidal product, we must pick something that is always true (like \(S\)) as our monoidal unit. If we had chosen â€śorâ€ť then weâ€™d have to come up with a statement that is always false.

An â€śandâ€ť operation is clearly already associative and commutative. Itâ€™s also order-preserving, which we can use in the following proof (in a style suggested by Example 1.123):