ece108: add cross product

docs/1b/ece108.md

@@ -395,6 +395,10 @@ $$
 \overline S=\mathcal U-S
 $$
 
+The intersection and union operators have the same properties as **AND** and **OR** and so are equally commutative / associative.
+
+**De Morgan's laws** still hold with sets.
+
 ### Intervals
 
 An interval can be represented as a bounded set.
@@ -402,3 +406,23 @@ An interval can be represented as a bounded set.
 $$[a,b)=\{x\in\mathcal U|a\leq x\wedge x<b\}$$
 
 $\empty$ is any impossible interval.
+
+## Ordered pairs
+
+!!! definition
+    - A **binary relation** on two sets $A, B$ is a subset of their Cartesian product.
+    - An ***n*-ary relation** between $n$ sets is a subset of their *n*-Cartesian product.
+
+Also known as **tuples**, ordered pairs are represented by angle brackets.
+
+$$\left<a,b\right> = \left<c,d\right>\iff (a=c)\wedge(b=d)$$
+
+The **Cartesian product** of two sets is the set of all ordered pair combinations within the two sets.
+
+$$A\times B=\{\left<a,b\right> | (a\in A)\wedge (b\in B)\}$$
+
+It is effectively the cross product, so is not commutative, although distributing unions, intersections, and differences works as expected.
+
+The **n-Cartesian product** of $n$ sets expands the Cartesian product.
+
+$$A\times B\times\dots\times Z=\{\left<a, b,\dots z\right>|a\in A, b\in B,\dots,z\in Z\}$$