Solutions to Exercises

Mathematical Writing - Vivaldi Franco 2014


Solutions to Exercises

Exercise 1.1

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise 1.2

1.

3.

5.

7.

9.

11.

13.

15.

17.

19.

21.

23.

25.

27.

29.

Exercise 2.1

BAD: Is 39 a prime number? [ Specific and insignificant. ]

GOOD: Why is 1 not a prime number?

BAD: What is $$1/2+1/2^2+1/2^3$$ ?

GOOD: Is a fraction the same as a rational number?

BAD: What is the real part of $$2+3\mathrm {i}$$ ?

GOOD: What is the real part of $$\mathrm {i}^{\mathrm {i}}$$ ?

Exercise 2.2

BAD: The set of natural numbers less than 10.

GOOD: The set of the proper subsets of a finite set.

Exercise 2.3

1.

3.

5.

7.

9.

Exercise 2.4

1.

3.

5.

7.

9.

Exercise 2.5

1.

3.

5.

7.

9.

Exercise 2.6

1.

3.

5.

7.

9.

Exercise 2.7

1.

Exercise 2.8

1.

Exercise 2.11 First fix some ambient set. What are the domain and co-domain of such a function?

Exercise 2.12 Consider infinite sequences of non-negative integers, where the $$k$$ th term represents the power of the $$k$$ th prime in the prime factorisation of an integer.

Exercise 2.13

1.

2.

3.

4.

5.

Exercise 3.1

1.

3.

5.

Exercise 3.2

1.

3.

5.

7.

9.

Exercise 3.3

1.

3.

5.

7.

9.

Exercise 3.4

1.

3.

5.

7.

9.

Exercise 3.5

1.

Exercise 4.1

1.

3.

Exercise 4.2

1.

3.

5.

7.

9.

Exercise 4.3 In part (i) there is freedom in the choice of the ambient set: the smaller the set, the simpler the predicate. (But if the set becomes too small the implication disappears, see Exercise 4.10 .3.)

1.

3.

5.

7.

9.

Exercise 4.4 This is the definition of nephew:

$$x$$ is a nephew of $$y\,:=\,$$ There is $$z$$ such that $$x$$ is a son of $$z$$ and

    ( $$z$$ is a brother of $$y$$ or $$z$$ is a sister of $$y$$ ).

Thus to define nephew we must first define of brother and sister; in turn these will require other definitions.

Exercise 4.5

(a.1)

(a.3)

(b.1)

(c)

Exercise 4.6

1.

Exercise 4.7 Use Theorem 4.3 , Sect. 4.5 .

1.

3.

5.

Exercise 4.9 Let $$G=\{1,\ldots ,n\}$$ . Think of the computation of the predicate as the evaluation of the entries of an $$n\times n$$ matrix, where the $$(i,j)$$ -entry is the value of ’ $$i$$ loves $$j$$ ’. Then decide the order in which the entries are accessed, e.g., first by row, then by column.

1.

Exercise 4.10

1.

3.

Exercise 5.1 First translate symbols into words literally; then synthesise the meaning of the literal sentence. (It may be helpful to draw the graph of a function that has the stated property, and a function that hasn’t.)

1.

3.

5.

7.

9.

Exercise 5.3

1.

3.

5.

7.

Exercise 5.4

1.

3.

5.

7.

Exercise 5.5

1.

3.

5.

Exercise 5.6

1.

3.

Exercise 5.8

1.

3.

5.

Exercise 5.10

1.

3.

5.

7.

Exercise 5.11

1.

3.

5.

7.

9.

Exercise 6.2

1.

2.

Exercise 6.5 Imagine that there are 100 boxes, instead of three; the presenter opens 98 boxes, all of them empty.

Exercise 6.7

1.

2.

3.

4.

Exercise 7.1

1.

3.

Exercise 7.2

(a) The statement of the theorem is imprecise in several respects.

·  The nature of the numbers $$x$$ and $$y$$ is not specified (the inequality would be meaningless for complex numbers).

·  The case in which one of $$x$$ or $$y$$ is zero should be excluded, since in this case the left-hand side of the inequality is undefined.

·  The statement is false unless the inequality is made non-strict; indeed the equality holds for infinitely many values of $$x$$ and $$y$$ .

There are several flaws in the proof.

·  The basic deduction is carried out in the wrong direction, which proves nothing. (Proving that $$P\Rightarrow $$ TRUE gives no information about $$P$$ .)

·  The assertion ’the last equation is trivially true’ is, in fact, false for $$ x=y$$ .

·  The writing is inadequate, without sufficient explanations, and also imprecise (the expression $$(x-y)^2>0$$ is an inequality, not an equation).

(b) Revised statement:

THEOREM:

For all nonzero real numbers $$x$$ and $$y$$ , the following holds:

$$ \frac{x^2+y^2}{|xy|}\geqslant 2. $$

PROOF:

Let $$x$$ and $$y$$ be real numbers, with $$xy\ne 0$$ . We shall deduce our result from the inequality $$(x\pm y)^2\geqslant 0$$ . We begin with the chain of implications:

$$ (x\pm y)^2\geqslant 0\Rightarrow x^2\pm 2xy+y^2\geqslant 0\Rightarrow x^2+y^2 \geqslant \mp \, 2xy. $$

Now, since $$xy$$ is non-zero, we divide both sides of the rightmost inequality by $$|xy|$$ , to obtain

$$ \frac{x^2+y^2}{|xy|}\geqslant \mp \, 2\frac{xy}{|xy|}. $$

The expression $$xy/|xy|$$ is equal to $$1$$ or $$-1$$ , and by choosing an appropriate sign we can ensure that $$\pm xy/|xy|=1$$ . Our proof is complete. $$\square $$

Exercise 7.3

2.

Exercise 7.4

1.

3.

5.

7.

9.

Exercise 7.5 Consider Euclid’s algorithm.

Exercise 8.1 The difficulty in a computer-assisted proof consists in converting the two sides of the identities ( 8.4 ) to the same form. In the following Maple codes the function expand is used for this purpose. In each case, the output of the last expression is the boolean constant TRUE .

3. The function sum performs symbolic summation.

5. The base case is $$n=2$$ .

Exercise 8.4

1.

3.

Exercise 8.5

1.

Exercise 9.1

1.

2.

3.

Exercise 9.2

1.

3.

Exercise 9.3

1.

Exercise 9.4

1.

2.

3.

Exercise 10.1

1.

2.

3.

4.

Exercise 10.3 How does $$k$$ depend on $$n$$ when $$n$$ is large? Consider Stirling’s formula 1 for the factorial.

Exercise 10.4

1.

2.

3.