2.6. Some Notes on Proofs and Abstraction 39
the whole contraption go up in flames when it runs into the statement
it contradicts.
Your audience is a person who is familiar with the underlying
definitions used in the statement being proved, but not the statement
itself. For instance, it could be yourself after you learned the defi-
nitions, but before you had begun work on the proof. You do not
have to put every tiny painful step in the write-up, but be careful
about what you assume of the reader’s ability to fill in gaps. Your
goal is to convince the reader of the truth of the statement, and that
requires the reader to understand the proof. Along those lines, it is
often helpful to insert small statements (I call it “foreshadowing” or
“telegraphing”) that let the reader know why you are doing what you
are currently doing, and where you intend to go with it. In particular,
when working by contradiction or induction, it is important to let the
reader know at the beginning. More complicated proofs, the kinds
that take several pages to complete, often benefit from an expository
section at the beginning, that outlines the proof with a focus on the
“why” of each step. A more technical portion that fills in all the
details comes afterward.
You must keep to the definitions and other statements as they are
written. In fact, a good strategy for finding a proof of a statement is
first to unwrap the definitions involved. Be especially wary of men-
tally adding words like only, for all, for every, or for some which are
not actually there. If you are asked to prove an implication it is likely
the converse does not hold, so if you “prove” equivalence you will be
in error. Statements that claim existence of an object satisfying cer-
tain hypotheses may be proved by producing an example, but if you
are asked to prove something holds of all objects of some type, you
cannot do so via a specific example. Instead, give a symbolic name
to an arbitrary object and prove the property holds using only facts
that are true for all objects of the given type. Similarly, the term
without loss of generality, or WLOG, appears from time to time in
proofs to indicate a simplifying but not restrictive assumption. If you
use the term make sure the assumption truly does not restrict the
cases. For example, one may assume without loss of generality that
the coefficient of x in the equation of a plane is nonnegative, since if
Previous Page Next Page