Homework Expectations
September 10, 2014
1. Let P be the statement ‘The sum of two odd numbers is odd’.
(a) Write the statement of P formally.
(b) Write the negation of P .
(c) Prove P , or disprove P and salvage it.
The solution to this problem must look like the following:
1. Let P be the statement ‘The sum of two odd numbers is odd’.
(a) Write the statement of P formally.
Solution for this part.
(b) Write the negation of P .
Solution for this part.
(c) Prove P , or disprove P and salvage it.
Solution for this part.
That is, you must state the problem immediately before the solution, or you
will lose points (this is because in mathematical writing, it is always important
to state the theorem before the proof, so that the reader knows what is about
to be shown).
Solutions for the first part could include ‘for every pair of numbers n and m,
if n is odd, and m is odd, then n + m is odd.’ Another acceptable solution is
’for every pair of odd numbers n and m, it is the case that n + m is odd’. Note
that quantifiers are required: without the quantifiers, the formal statement is a
dependent statement, while the original statement was not.
Of course, sometimes it is okay to leave off a quantifier. For instance, if the
problem was ‘If f (x) is continuous on [a, b], then f (x) is integrable on [a, b].’ In
this example, the informal statement involves the function f (x), and the interval
[a, b], and hence is a dependent statement, and does not need quantifiers (if this
fact is confusing, you may still choose to include quantifiers, and you will not lose
points. However, it is only necessary to put quantifiers for informal statements
that appear to not be dependent statements).
The negation would be ‘there exists a pair of numbers n and m, such that
n is odd, m is odd, and n + m is even’.
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For the last part, we note that P is false. So I would expect to see the
statement ‘P is false, because if n = 3, and m = 5, then n + m = 8, which is not
odd.’ This answer would get full points whenever P is false, and the statement
of the problem was to prove or disprove P .
However, when I say disprove and salvage, this means you need to replace
P with an obvious statement Q that is true. In this case, our example suggests
that the sum of two odd numbers is even, so I would expect you to say so, and
then to prove it.
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Proving Results
Now we discuss how to write proofs. Recall of course, from above, that you
need to state what you are proving first. You also need to start the proof with
the word Proof, followed by a colon (:). This should be left justified. Then you
begin with the first sentence, capitalized. You must always write in complete
sentences. However, as noted in class, the sentence ‘Then equation 1 = equation
2 = ... = final equation’ counts as a correct sentence, as long as each equation
only differs from the previous by a simple calculuation. Also, as was seen in
class, it is customary for such a sentence to look like this:
Then
2q + 1 + 2r + 1 = 2q + 2r + 1 + 1
=
2q + 2r + 2
=
2(q + r + 1).
Sometimes (although probably not in this course), one of the equalities arises
from an application of a theorem (for instance, Euler’s formula). Then it is
customary to end the set of equations with a comma instead of a period, and
then to include the words ‘where equality (x) follows from Euler’s formulas.’
Remember that you should not use variables without introducing them. Also,
it is good manners to make it clear when you are applying a definition, or a
theorem. Later in life you will be allowed to omit sentences. However, for this
class, the following is expected:
1. If you are applying a definition, which uses an existential quantifier, there
should be a sentence such as: ‘Since n is odd, there exists an integer q
such that n = 2q + 1. You need to refer both to the fact that n is odd,
and to the fact that q is an integer!
2. Sometimes you will apply a definition involving a universal quantifier. I
will cover this when the time arises.
3. Sometimes you will apply a Theorem - I will cover what to do in this case
as well.
For the first homework, typically you will only be applying definitions involving
existential quantifiers. Also, there is one exception to my above rule. Suppose
you were to apply the definition of odd to two different integers m and n. Then
I allow the first two approaches:
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1. Since n and m are both odd, there exists integers q and r, such that
n = 2q + 1 and m = 2r + 1.
2. Since n is odd, there exists and integer q such that n = 2q + 1. Similarly,
since m is odd, there exists and integer r such that n = 2r + 1.
3. The following approach is unacceptable: Since n is odd, there exists and
integer q such that n = 2q + 1. Similarly, m = 2r + 1 for some r.
Note that, every time you apply a quantifier, you should use different variable
names! Otherwise the proof might not be correct (it might end up implying that
n = m in the above example, for instance).
Finally, please note that this is a writing class. Turned in homework should
be viewed as being a final draft. Thus, you will lose points for having crossed out
sentences, or if there are obvious grammatical mistakes. Mathematicians also
prefer their proofs to be concise, without including comments and calculations
in a proof that are not absolutely necessary. If you have a sentence that could
be removed, or there is a way to shorten your answer, it will be noted on graded
homework. However, you will not lose points for these issues, nor for spelling
mistakes.
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