Exercise 2.35

Exercise 2.35#


It should satisfy the identity/unitality condition because \(S \cap X\) for any \(X\) that is a subset of \(S\) should be \(X\). See also Intersection (set theory); the intersection operation is associative, and commutative.

To see the monotonicity condition is satisfied we’ll use as an inference rule that a partially-applied \(\cap\) is order-preserving. That is, if we have that \(a_1 \leq a_2\) (that is, \(a_1 \subset a_2\)), it follows that \(a_1 \cap b \leq a_2 \cap b\). Then, starting from these two premises:

\[\begin{split} \begin{align} \\ x_1 & \leq y_1 \\ x_2 & \leq y_2 \\ \end{align} \end{split}\]

Applying \(\_ \cap y_2\) to the first premise:

\[\begin{split} \begin{equation} \\ x_1 \cap y_2 \leq y_1 \cap y_2 \\ \label{eq:35a} \tag{a} \end{equation} \end{split}\]

Applying \(x_1 \cap \_\) to the second premise:

\[\begin{split} \begin{equation} \\ x_1 \cap x_2 \leq x_1 \cap y_2 \\ \label{eq:35b} \tag{b} \end{equation} \end{split}\]

And combining \(\eqref{eq:35a}\) and \(\eqref{eq:35b}\) with transitivity:

\[ x_1 \cap x_2 \leq x_1 \cap y_2 \leq y_1 \cap y_2 \]