# The Universe of Discourse

Mon, 01 Nov 2021

!!\def\zpr#1#2{\langle{#1},{#2}\rangle}\def\zkp#1#2{\{\{{#1}\}, \{{#1},{#2}\}\}}!!In Friday's article about the cartesian product I needed to show what the Kuratowski ordered-pair construction looks like when you nest the pairs. I originally wrote out the TeX for these by hand, but later decided I ought to use TeX macros to generate the formulas. First I did

    \def\pr#1#2{\langle{#1},{#2}\rangle}


so that \pr ab would turn into !!\zpr ab!! and \pr a{\pr bc} would turn into !!\zpr a{\zpr bc}!!.

Then I defined a macro for Kuratowski pairs. The Kuratowski pair for !!\zpr ab!! is the set !!\zkp ab!!, which is kind of a mess:

    \def\kp#1#2{\{\{{#1}\}, \{{#1},{#2}\}\}}


Then the nested Kuratowski pairs turn into:

$$\begin{array}{cc} \verb+\kp a{\kp bc}+ & \zkp a{\zkp bc} \\ \verb+\kp{\kp ab}c+ & \zkp{\zkp ab}c \\ \end{array}$$

When I got this far I realized that my hand-expansion of \kp{\kp ab}c had actually been wrong! I had originally written: $$\{\{\{a\}, \{a, b\}\}, \{\{\{a\}, \{a, b\}\}, c\}\}.\qquad\color{\maroon}{Wrong!}$$

(There's a pair of braces missing around the first of the two !!\zkp ab!!.)

I used to give classes on programming style and technique, and one of the maxims I taught was “let the computer do the work”: use the computer to automate repetitive or error-prone tasks.

I was going to say I wish I'd taken my own advice here but hey — I did take my own advice, and it worked!