These notes accompany the book Surreal Numbers by Donald Knuth, specifically
the
1974 edition. They are only intended to further my own understanding; no
guarantees of accuracy, relevance, or significance are extended.
A surreal number $x$ consisting of left set $X_L$ and right set $X_R$ is
represented as
\surreal{X_L
}{X_R
}. The void set, as Knuth named it, is
represented by leaving the appropriate left or right set empty, as in
\surreal{}{}, the first surreal number defined.
When applying binary relations like less-than-or-equal to sets, the notation $X
\leq Y$ means that, $
\forall x
\in X$ and $
\forall y
\in Y$, it holds true that