Math 300A: Rules for Citing Reasons in Proofs
Remember that there is an "Imagined Reader" for your proofs in
homework to whom you have to explain your reasoning.
We assume that this reader has a copy of
Axioms for the Real Numbers, but hasn't memorized it.
So we have to explicitly tell the reader when we use a definition, axiom,
or theorem with a lower number than the one we are proving.
Notes below tell you how much more you may assume about your Imagined Reader
for each homework.
First homework due 10/4, repeat of what it says in the pdf:
You must mention every axiom, definition, and previous theorem that you use
in your reasoning, and also must mention when you use substitution.
(The other three properties of equality - reflexivity, symmetry, and
transitivity - do not have to mentioned when you use them. But you may cite
them if you need to; for instance, in a 2-column proof you might have
a step in which one of these properties is the only reason used.)
- Second homework due 10/11: Our Imagined Reader now knows
Axioms 1-4 (Closure and the Commutative, Associative, and Distributive Laws)
well enough that you do not need to refer to them every time you use them.
Also our reader know the definitions of difference and quotient, and can
combine them with the axioms and properties of equality.
This means you do not need to give reasons why you may replace, for instance,
"(a + b) - c" by "(b - c) + a"
(which otherwise would require using definition of difference, Axiom 3,
Axiom 2, and the difference definition again).
And we can drop parentheses when we add or multiply several numbers,
becase our reader knows why the order doesn't matter.
Don't forget to say when you use substitution.
- Third homework due 10/18: Now our Imagined Reader is very
comfortable with Axioms 1-7 and Theorems 1-38, so you do not need to cite
any of these to justify your algebraic manipulations. Also our reader
doesn't need you to prove that specific numbers are odd or even; e.g.,
you may say that 2 is even or that -5 is odd without proving it from the
defintion of even or odd.
Be sure to read all the Practice Problems.
For instance, §1.2: 1, 2, and 3 ask you to do eight 2C proofs.
After doing a couple of them, you will probably be confident
you could do the rest, and don't need to work out all of them.
But they are all listed as Practice Problems so you can cite them as reasons:
you may use a result from a Practice Problem as a
reason for a step in a proof of a later problem.
- Rest of the quarter: The Imagined Reader now knows all the
results in ``Axioms for Real Numbers.'' You still need to verify
hypotheses. For example, if you multiply an inequality by a number c,
you need to state whether c is positive, negative, or zero, and
say what effect this has on the equality; but you don't have to give a theorem
(However, if it was something you had to look up to be sure it was right,
then do cite the theorem you checked!)
You may state results on parity (even and odd) relating the parity of
two numbers to the parity of their sum and product without citing a theorem,
problem, or page number. This includes the result that an integer
n and its square have the same parity.
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Most recently updated on October 23, 2017.