Department of Mathematics and Applied Mathematics
MAM1019H
Tutorial 7
2024
These are questions on Chapter 6 (Proof by contradiction) and Sections 11.1 and 11.2
(Relations) of the textbook, as well as a few more questions on the properties of real
numbers. Questions 1 to 5 are on proof techniques; Question 6 on relations. Please make
sure you do at least some of the questions on relations before the lectures next week (the
week starting 22 April).
1. You remember that we showed that the sum and the product of any two rational
numbers is rational. Prove that the corresponding results for irrational numbers
fail. Do so by providing two irrationals whose sum is rational, and two irrationals
whose product is rational.
(This is straightforward: you can do it in your head.)
2. Read the following theorem and the outline of the proof given. Then answer the
questions below. The idea is that you understand every single step in the argument,
by the time you’ve finished! Being able to read and understand new proofs is important; spend as much time as you need on this.
Theorem
There do not exist prime numbers a, b and c such that a3 + b3 = c3 .
Proof We use proof by contradiction.
So assume there exist prime numbers a, b and c such that a3 + b3 = c3 .
(1)
(2)
If a and b are both odd, then a3 + b3 is even.
So c3 is even.
So c is even.
So c = 2.
This is not possible, since a and b, being odd primes, are both greater than 2.
(3)
(4)
(5)
(6)
(7)
So at least one of a and b is even. Say it’s b.
Then b = 2.
Then b3 = 8 = c3 − a3 = (c − a)(c2 + ca + a2 ).
Now each of c2 , ca and a2 is greater than or equal to 4.
So c2 + ca + a2 ≥ 12.
So c3 − a3 ≥ 12.
This is a contradiction, which concludes the proof.
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(a) Show that a3 + b3 is even, as claimed in line (3).
(b) Explain why c is even, as claimed in line (5).
(c) In line (6), it is claimed that c = 2. Why can’t c be some other even number?
(d) The argument in lines (3) to (7) is also a proof by contradiction. To prove a
statement P , one assumes ∼ P and gets a contradiction. What are ∼ P and
P , in this case?
(e) Check that you agree with the equations in line (10).
(f) Explain the claim in line (11).
(g) Explain the claim in line (12).
(h) To get from line (12) to line (13), one uses c − a ≥ 1. Why does this hold?
(i) What is the contradiction mentioned in line (14)?
3. In lectures, we proved that the cube root of 5 is irrational. Go through the proof
and explain why the same proof cannot be used to show that the cube root of 8 is
irrational.
√
4. Prove or disprove: 15 is irrational.
5. Chapter 6. Do a selection from questions 7, 11, 13, 15, 17 on page 144.
6. Chapter 11. Do a selection from section 11.1 on p204, and section 11.2 on pp208,209.