Concrete mathematics : a foundation for computer science

432 ASYMPTOTICS
subtle consideration lurks in the background. Namely, we need to realize that
it’s fine to write
in3 + in2 + An = O(n3),
but we should neueT write this equality with the sides reversed. Otherwise
we could deduce ridiculous things like n = n2 from the identities n = 0 (n2)
and n2 = O(n2). When we work with O-notation and any other formulas
that involve imprecisely specified quantities, we are dealing with one-way
equalities. The right side of an equation does not give more information than
the left side, and it may give less; the right is a “crudification” of the left.
From a strictly formal point of view, the notation 0( g(n)) does not
stand for a single function -f(n), but for the set of all functions f(n) such
that If(n)1 6 Clg(n)l for some constant C. An ordinary formula g(n) that
doesn’t involve O-notation stands for the set containing a single function
f(n) = g(n). If S and T are sets of functions of n, the notation S + T stands
for the set of all functions of the form f(n) + g(n), where f(n) E S and
g(n) E T; other notations like S-T, ST, S/T, &, es, In S are defined similarly.
Then an “equation” between such sets of functions is, strictly speaking, a set
inclusion; the ‘=’ sign reall:y means ‘g’. These formal definitions put all of
our 0 manipulations on firm logical ground.
For example, the “equation”
in3 + O(n’) = O(n3)
means that S1 & S2, where S-I is the set of all functions of the form in3+f1 (n)
such that there exists a constant Cl with If,(n)/ 6 Clln’I, and where S2
is the set of all functions f.!(n) such that there exists a constant CJ with
Ifz(n)l 6 C21n31. we can formally prove this “equation” by taking an arbitrary
element of the left-hand side and showing that it belongs to the righthand
side: Given in3 + fl In) such that If,(n)1 < C11n21, we must prove
that there’s a constant Cl such that l$n3 + fl (n)l 6 C21n31. The constant
Cl = 3 + Cl does the trick, Isince n2 6 In31 for all integers n.
If ‘=’ really means ‘C’, why don’t we use ‘c’ instead of abusing the equals
sign? There are four reasons.
First, tradition. Number theorists started using the equals sign with Onotation
and the practice stuck. It’s sufficiently well established by now that
we cannot hope to get the mathematical community to change.
Second, tradition. Computer people are quite used to seeing equals signs
abused- for years FORTRAN and BASIC programmers have been writing
assignment statements like “N = N + 1’. One more abuse isn’t much.
Third, tradition. We often read ‘=’ as the word ‘is’. For instance we
verbalize the formula H, = O(log n) by saying “H sub n is Big Oh of log n!’
“And to auoide the
tediouse repetition
of these woordes:
is equal/e to: I will
sette as I doe often
in woorke use, a
paire of paralleles,
or Gemowe lines of
one lengthe, thus:
= , bicause
noe .2. thynges, can
be moare equal/e.”
-R. Recorde 12461