Concrete mathematics : a foundation for computer science

72 INTEGER FUNCTIONS
An important special case of this theorem is worth noting explicitly:
if m and n are integers and the denominator n is positive. For example, let
m = 0; we have [l[x/lO]/lOJ /lOI = [x/1000]. Dividing thrice by 10 and
throwing off digits is the same as dividing by 1000 and tossing the remainder.
Let’s try now to prove or disprove another statement:
This works when x = 7~ and x = e, but it fails when x = 4; so we know that
it isn’t true in general.
Before going any further, let’s digress a minute to discuss different “levels”
of questions that can be asked in books about mathematics:
Level 1. Given an explicit object x and an explicit property P(x), prove that
P(x) is true. For example, “Prove that 1x1 = 3.” Here the problem involves
finding a proof of some purported fact.
Level 2. Given an explicit set X and an explicit property P(x), prove that
P(x) is true for all x E X. For example, “Prove that 1x1 < x for all real x.”
Again the problem involves finding a proof, but the proof this time must be
general. We’re doing algebra, not just arithmetic.
Level 3. Given an explicit set X and an explicit property P(x), prove or
disprove that P(x) is true for all x E X. For example, “Prove or disprove In my other texts
that [ml = [J;;] for all real x 2 0.” Here there’s an additional level ~~se~~~~nr($
of uncertainty; the outcome might go either way. This is closer to the real Same as ~~~~~~~~~
situation a mathematician constantly faces: Assertions that get into books about 99.44% df
tend to be true, but new things have to be looked at with a jaundiced eye. If
the statement is false, our job is to find a counterexample. If the statement
is true, we must find a proof as in level 2.
the time; but not
in this book.
Level 4. Given an explicit set X and an explicit property P(x), find a necessary
and suficient condition Q(x) that P(x) is true. For example, “Find a
necessary and sufficient condition that 1x1 3 [xl .” The problem is to find Q
such that P(x) M Q(x). Of course, there’s always a trivial answer; we can
take Q(x) = P(x). But the implied requirement is to find a condition that’s as
simple as possible. Creativity is required to discover a simple condition that But no simpler.
will work. (For example, in this case, “lx] 3 [xl H x is an integer.“) The
extra element of discovery needed to find Q(x) makes this sort of problem
more difficult, but it’s more typical of what mathematicians must do in the
“real world!’ Finally, of course, a proof must be given that P(x) is true if and
only if Q(x) is true.
-A. Einstein