Student’s Manual Essential Mathematics for Economic Analysis 5th edition Knut Sydsæter

Student’s Manual

Essential Mathematics for Economic Analysis
5th edition

Knut Sydsæter Peter Hammond Andr´es Carvajal Arne Strøm

1          Essentials of Logic and Set Theory

Review exercises for Chapter 1

3.   Consider the Venn diagram for three sets depicted in Fig. SM1.R.3. Let n_k denote the number of students in the set marked (k), for k = 1, 2, . . . , 8. Suppose the sets A, B, and C refer to those who study English, French, and Spanish, respectively. Since 10 students take all three languages, n₇  = 10.  There are 15 who take French and Spanish, so 15 = n₂ + n₇, and thus n₂ = 5. Furthermore, 32 = n₃ + n₇, so n₃ = 22. Also, 110 = n₁ + n₇, so n₁ = 100.  The rest of the information implies that 52 = n₂ + n₃ + n₆ + n₇, so n₆ = 52 − 5 − 22 − 10 = 15. Moreover, 220 = n₁ + n₂ + n₅ + n₇, so n₅ = 220 − 100 − 5 − 10 = 105. Finally, 780 = n₁ + n₃ + n₄ + n₇, so n₄ = 780 − 100 − 22 − 10 = 648. The answers are therefore:

(a) n₁ = 100,

(b) n₃ + n₄ = 648 + 22 = 670,

i=1

(c) 1000 − Σ7          n_i = 1000 − 905 = 95.

Figure SM1.R.3

4.   (a) ⇒ is true; ⇐ is false, because x = y = 1 also solves x + y = 2.

(b)    ⇒ is false, because x² = 16 also has the solution x = −4; ⇐ true, because if x = 4, then

x² = 16.

(c)    ⇒ is true, because (x − 3)² ≥ 0; ⇐ false because with y  > −2 and x = 3, one has (x − 3)²(y + 2) = 0.

(d)    ⇒ and ⇐ are both true, since the equation x³ = 8 has the solution x = 2 and no others.¹

5.   For (a) and (b) see the solutions in the book. For (c), note that when n = 1, the inequality is obviously correct.² As the induction hypothesis when n equals the arbitrary natural number k, suppose that (1 + x)^k ≥ 1 + kx. Because 1 + x ≥ 0, we then have

(1 + x)^k+1 = (1 + x)^k(1 + x) ≥ (1 + kx)(1 + x) = 1 + (k + 1)x + kx² ≥ 1 + (k + 1)x.

where the last inequality holds because k > 0. Thus, the induction hypothesis holds for

n = k + 1. Therefore, by induction, Bernoulli’s inequality is true for all natural numbers n.

¹ In the terminology of Section 6.3, function f (x) = x³ is strictly increasing. See Fig. 4.3.7 and Exercise 6.3.3.

² And for n = 2, it is correct by part (b).