UMBC CMSC203, Discrete Structures, Spring 2009
Proofs are written in English
Do not worry overly much about the names of these rules of inference.
For this class, if you use the inferences correctly, that is good
enough. You don't have to remember which one is modus ponens
and which one is modus tollens. In fact, if you flip through
almost any math book, you will not find any mention of "modus ponens"
or "disjunctive syllogism" in the proofs (with the possible exception
of books on logic). This is because applying a rule of inference
is such a small step in the reasoning process that it would not be
helpful to the reader to point out that you have just used a
Mathematical proofs are usually written in paragraph form and complete
sentences. Even a sentence that uses mathematical notation should be
a complete sentence when read aloud. For example:
If the vertex x ∈ A, then we color x red.
This sentence should be read aloud as "If the vertex x is
an element of A, then we color x red." The verb in
the "if" clause is the ∈ symbol.
Most of the rules for writing you learned in English classes are
applicable in mathematical writing. For example, when you start a
new topic, you should start a new paragraph. You should also avoid
run-on sentences, they are hard to decipher. About the only exception
is the advice that you vary the vocabulary. You might have been
told in a writing class to use different words to describe an object
so you do not repeatedly use the same word. In mathematical writing,
you should actually stick to the same terminology. For example,
although the terms "vertex" and "node" are used interchangeably,
you should stick to one or the other and not use both words in the same
writing. Thus, in mathematical writing, you do not need to wrack
your brains to think up "exciting" adjectives and adverbs. For
example, a vertex without any edges is called an "isolated" vertex.
You will always call this an isolated vertex. You won't ever have to
describe the vertex as "lonely", "alienated", "dejected" or "outcast".
Instead, concentrate on the 4 C's of mathematical writing: try to be
clear, concise, convincing and correct.
- Being clear means that the reader understands your intended
meaning. It is helpful to give names to the objects you are talking
about and to use the names consistently. It is also important to
use mathematical notation correctly.
- Being concise means getting to the point. Don't introduce things that
are not relevant to the proof. If the proof is complicated, break it down
into smaller steps. Try to consolidate cases so there are not too many of
- Being convincing to a semi-skeptical reader is the main point of
writing a proof down. Proofs often have a key idea and the rest is just
"window dressing" to set up the proof. Make sure that this key
idea is not lost in the window dressing. If there are subtle points in the
proof, warn the reader. Try to anticipate any questions or objections that
the reader may have and answer them.
- Being correct is of course the final and most important point. A
well-written proof that is clear, concise and convincing is of no use if
the proof is actually wrong! Read and re-read your proof to look for bugs.
17 Sep 2013 22:24:26 EDT
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