Seminar 1: Definitions, axioms, direct proof, counter-examples.
Part 1: We begin with considering some di1erent types of mathematical statements
that we encounter in this course. For each of the statements (a) – (j) below, try to
address the following questions:
•
•
•
•
•
What type of statement is it? Definition, axiom, theorem, proposition?
Do all terms in the statement make sense to you?
Does it make sense to say that the statement is true or false?
How can we determine if the statement is true or false?
If it makes sense, determine whether the statement is true or false.
a)
c)
d)
e)
The natural numbers are those, starting from 1, that you get by adding 1
with itself.
We call a natural number larger than 2 a prime number if it can only
divided by 1 or itself.
We call all natural numbers divisible by 2 an even number.
We call all natural numbers larger than 0 positive.
All even natural numbers are prime numbers.
f)
g)
h)
i)
j)
For all real numbers x,y it is true that xy = yx.
For all real numbers x it is true that (x+2)^2 = x^2 + 2^2.
For some real numbers x it is true that (x+2)^2 = x^2 + 2^2.
For all non-zero real numbers a, b it is true that 1/(a+b) = 1/a + 1/b.
For some non-zero real numbers a, b it is true that 1/(a+b) = 1/a + 1/b.
b)
Challenge:
(-1)(-1) = 1
Part 2: When working on so-called “complex fractions” – that is, fractions of fractions
such as (a/b)/(c/d) – then the technique of “multiplying by 1” is very useful (it is also
called “creative 1’s).
Note: The technique is explain in Example 3.6 in Module 3 of MNXA21.
(a) Use creative 1’s to express (3/10)/(6/5) as a “simple fraction” (that is, on the form
x/y).
(b) Use creative 1’s to express (a/b)/(c/d) as a simple fraction. Do not worry about
axioms and such – just make sure each step seems reasonable.
(c) Do any of your ‘reasonable’ steps match any of those on pages 4 and 5 in Don’t
Panic?
Part 3: We now look more closely at how to multiply out the expression (a+b)(c+d+e).
(a)
(b)
Use an algebraic argument to simplify the above expression by multiplying out
the parentheses (do not worry about axioms for the moment – just do as you
would in high school).
A visual argument for the formula you found in (a) is given in Proposition 3.4 of
Module 3 of MNXA21. Which of these two arguments is the most correct?
Which is the most convincing? What argument do you find the most
“pedagogical” in the sense that you can use it to convince someone?
Part 4: We now look at di1erent ways to prove Gauss’ summation formula for the sum
1+2+3+4+…+n.
This summation formula can be proved in a million di1erent ways (I think), and was
apparently first found by Gauss when he was 6 year old (Gauss later grew up to become
one of the greatest – if not ‘the’ greatest - mathematicians of all time).
(a) The idea of 6-year old Gauss was to write up the numbers in the way shown
below. Illustrated using the case n = 8, the inspiration of Gauss’ came from
writing these numbers in the following way.
Do you see how to use this to quickly compute the sum?
(b) Try to extend Gauss’ idea from n=6 to a general n and use this to work out what
his formula is. (It may be an advantage to start by assuming that n is even.)
(c) Check if the formula you found in (b) matches what you found in (a). What does
this tell you?
(d) Another proof can be found by thinking of the sum in term of adding up areas of
1x1 squares places in a clever way so that one can use formulas for the area of a
triangle. Try to see if you can come up with such a way.
(e) In the textbook, there is a more “formal” proof by induction shown in Example
1.80 on page 31. Take a quick (and superficial) look at that proof. Do you find it
convincing? Do you find that it gives you insight? (We return to this proof in the
lectures).