## 2014-01-15

### Set Differences

Off-day in the AM. Planning to hit the cafe to study 1) set differences, 2) MathJax's indentation limitations, 3) encoding propositions that are machine-readable, for popular automated theorem solvers.

As if there are "popular" automated theorem solvers. Sometimes I look at myself and wonder if I'm crazy. (Almost everyone else already seems sure that I am.)

9pm update: 1) and 2) addressed. A number of irrelevant distractions from errands and accidental participation in a staff meeting seem to have exhausted my ability to study. Time for a break.

Theorem:
\begin{align*}
(\text{A is a set}" \wedge
\text{B is a set}" \wedge
\text{C is a set}") \implies \\
&C \backslash(A\cup B) =(C\backslash A)\cap(C\backslash B)
\end{align*}

Proof:
\begin{align*}
x\in C \backslash(A\cup B)
&= \underbrace{x\in C}\wedge\underbrace{x\notin (A\cup B)}\\
&= \underbrace{x\in C}\wedge\underbrace{(x\notin A\wedge x\notin B)}\\
&= \underbrace{(x\in C \wedge x\notin A)}\wedge \underbrace{(x\in C \wedge x\notin B)}\\
&= \underbrace{x\in (C\backslash A)} \wedge \underbrace{x\in (C\backslash B)}\\
&=x\in \underbrace{(C\backslash A)\cap(C\backslash B)}
\end{align*}
"Are you the mathematician? I heard that this place has a mathematician."
"Er, well, I'm studying math, but I'm not sure if that makes me a mathematician. Today I'm reading about 'power sets' - ask me in a coupla years what those are for."

:P