Solutions 1 - Speaking Mathematically

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Exercise set 1.1

Question 1

(a) $x^{2} = - 1$

(b) a real number $x$

Question 2

(a) a remainder of $2$ when it’s divided by $5$ and a remainder of $3$ when it’s divided by $6$

(b) an integer $n$ ; $n$ is divided by $6$ the remainder is $3$

Question 3

(a) $a < c < b$ if $a < b$ , or $b < c < a$ if $b < a$

(b) distinct real numbers, $a$ and $b$

Question 4

(a) a real number; $s > r$

(b) real number $r$ , there is a real number $s$

Question 5

(a) $r$ is positive

(b) positive; the reciprocal of $r$ is also positive

(c) is positive; the reciprocal of $r$ is also positive

Question 6

(a) $s$ is negative

(b) negative; the cube root of $s$ is also negative

(c) is negative; the cube root of $s$ is also negative

Question 7

(a) There are real numbers such that their sum is less than their difference.

True; $u = - 1, v = - 2 ⟹ u + v = - 3 < u - v = 1$

(b) There is a real number such that it’s greater than its square.

True; $x = \frac{1}{2} ⟹ x^{2} = \frac{1}{4} < x = \frac{1}{2}$

(c) For every positive integer, the square of that integer is greater than the integer itself.

False; $x = \frac{1}{2} ⟹ x^{2} = \frac{1}{4} < x = \frac{1}{2}$

(d) For every two real numbers, the absolute value of their sum is less than or equal to the sum of their absolute values.

True

Question 8

(a) have four sides

(b) has four sides

(c) has four sides

(d) is a square; has four sides

(e) $J$ has four sides

Question 9

(a) have at most two real solutions

(b) has at most two real solutions

(c) has at most two real solutions

(d) is a quadratic equation; has at most two real solutions

(e) $E$ has at most two real solutions

Question 10

(a) have a reciprocal

(b) a reciprocal

(c) $s$ is the reciprocal of $r$

Question 11

(a) have a positive square root

(b) positive square root

(c) $r$ is the positive square root of $e$

Question 12

(a) real numbers; product with every number leaves the number unchanged

(b) with every number leaves the number unchanged

(c) $rs = s$

Question 13

(a) real number; product with every real number equals zero

(b) with every real number equals zero

(c) $ab = 0$

Exercise set 1.2

Question 1

A = C, B = D

Question 2

(a) The set of all real positive numbers $x$ such that $x$ is between $0$ and $1$ .

(b) The set of all real numbers $x$ , such that $x$ is less than or equal to $0$ or $x$ is greater than or equal to $1$ .

(c) The set of all integers $n$ , such that $n$ is a factor of $6$ .

(d) the set of all positive integers $n$ , such that $n$ is a factor of $6$ .

Question 3

(a) No.

(b) 3.

(c) 3.

Question 4

(a) Yes.

(b) One.

(c) Two.

(d) Yes.

(e) No.

Question 5

Only $D$ and $A$ are equal.

Question 6

$T_{- 3} = {- 3, 9} ⟹ ∣ T_{- 3} ∣ = 2$
$T_{0} = {0} ⟹ ∣ T_{0} ∣ = 1$
$T_{1} = {1} ⟹ ∣ T_{1} ∣ = 1$
$T_{2} = {2, 4} ⟹ ∣ T_{2} ∣ = 2$

Question 7

(a) $S = {- 1, 1}$

(b) $T = {0, 2}$

(c) $U = {- 2, - 1, 0, 1, 2}$

(d) $V = {\dots, - 1, 0, 1, \dots} = Z$

(e) $W = {} = ϕ$

(f) $X = {\dots, - 1, 0, 1, \dots} = Z$

Question 8

(a) False $j \in B, j \in A$

(b) True

(c) True

(d) True

Question 9

(a) Yes

(b) No

(c) No

(d) Yes

(e) Yes

(f) No

(g) Yes

(h) No

(i) Yes

(j) Yes

Question 10

(a) No, $(4, - 4) \neq = (- 4, 4)$

(b) No

(c) Yes

(d) Yes

Question 11

(a) $A \times B = {(w, a), (w, b), (x, a), (x, b), (y, a), (y, b), (z, a), (z, b)}$

(b) $B \times A = {(a, w), (a, x), (a, y), (a, z), (b, w), (b, x), (b, y), (b, z)}$

(c) $A \times A = {(w, w), (w, x), (w, y), (w, z), (x, w), (x, x), (x, y), (x, z), (y, w), (y, x), (y, y), (y, z), (z, w), (z, x), (z, y), (z, z)}$

(d) $B \times B = {(a, a), (a, b), (b, a), (b, b)}$

Question 12

(a) $S \times T = {(2, 1), (2, 3), (2, 5), (4, 1), (4, 3), (4, 5), (6, 1), (6, 3), (6, 5)}$

(b) $T \times S = {(1, 2), (1, 4), (1, 6), (3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6)}$

(c) $S \times S = {(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6)}$

(d) $T \times T = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)}$

Question 13

(a) $A \times (B \times C) = {(1, (u, m)), (1, (u, n)), (2, (u, m)), (2, (u, n)), (3, (u, m)), (3, (u, n))}$

(b) $(A \times B) \times C = {((1, u), m), ((2, u), m), ((3, u), m), ((1, u), n), ((2, u), n), ((3, u), n)}$

(c) $A \times B \times C = {(1, u, m), (1, u, n), (2, u, m), (2, u, n), (3, u, m), (3, u, n)}$

Question 14

(a) $R \times (S \times T) = {(a, (x, p)), (a, (x, q)), (a, (x, r)), (a, (y, p)), (a, (y, q)), (a, (y, r))}$

(b) $(R \times S) \times T = {((a, x), p), ((a, x), q), ((a, x), r), ((a, y), p), ((a, y), q), ((a, y), r)}$

(c) $R \times S \times T = {(a, x, p), (a, x, q), (a, x, r), (a, y, p), (a, y, q), (a, y, r)}$

Question 15

{0000, 1000, 0100, 0010, 0001}

Question 16

{xxxx y, xxx y x, xx y xx, x y xxx, y xxxx}

Exercise set 1.3

Question 1

(a) $4 R 6; 4 R 8; (3, 8) \in R; (2, 10) \in R$

(b) $R = {(2, 6), (2, 8), (2, 10), (3, 6), (4, 8)}$

(c) $domain: {2, 3, 4}; co-domain: {6, 8, 10}$

Question 2

(a) $2 S 2; - 1 S - 1; (3, 3) \in S; (3, - 3) \in S$

(b) $S = {(- 3, - 3), (- 2, - 2), (- 2, 2), (- 1, - 1), (- 1, 1), (1, - 1), (1, 1), (2, - 2), (2, 2), (3, 3)}$

(c) $domain: {- 3, - 2, - 1, 1, 2, 3}; co-domain {- 3, - 2, - 1, 1, 2, 3}$

Question 3

(a) $3 T 0; 1 T - 1; (2, - 1) \in T; (3, - 2) \in T$

(b) $T = {(1, - 2), (2, - 1), (3, 0)}$

(c) $domain: {1, 2, 3}; co-domain: {- 2, - 1, 0}$

Question 4

(a) $2 V 6; (- 2) V 8; (0, 6) \in V; (2, 4) \in V$

(b) $V = {(- 2, 6), (0, 4), (0, 8), (2, 6)}$

(c) $domain: {- 2, 0, 2}; co-domain: {4, 6, 8}$

Question 5

(a) $(2, 1) \in S; (2, 2) \in S; 2 S 3; (- 1) S (- 2)$

Question 6

(a) $(2, 4) \in R; (4, 2) \in R; (- 3) R 9; 9 R (- 3)$

Question 7

(b)

$R$ is not a function since $(6, 5), (6, 6) \in R$
$S$ is not a function since $(5, 5), (5, 7) \in S$
$T$ is not a function since $(6, 5), (6, 7) \in T$

Question 8

(b)

$U$ is not a function because there is no $y \in B$ that satisfies $(2, y) \in U$
$V$ is not a function because there is no $y \in B$ that satisfies $(2, y) \in V$
$W$ is not a function since $(2, 5), (2, 3) \in W$

Question 9

(a) There is only one function: $F = {(0, 1), (1, 1)}$

(b) For every $(x, y) \in A \times B$ :

$(x, y) \in U$ means that $y = x$
$(x, y) \in V$ means that $y > x$

Question 10

$A = {(a, x), (a, y)}$
$B = {(b, x), (b, y)}$
$C = {(a, x)}$
$D = {(b, x)}$

Question 11

$L (0201) = 4$
$L (12) = 2$

Question 12

$C (x) = y x$
$C (yy x y x) = yyy x y x$

Question 13

(a) $domain: {- 1, 0, 1}; co-domain: {t, u, v, w}$

(b) $F (- 1) = u; F (0) = w; F (1) = u$

Question 14

(a) $domain: {1, 2, 3, 4}; co-domain: {a, b, c, d}$

(b) $G (1) = G (2) = G (3) = G (4) = c$

Question 15

Only d

Question 16

f (- 1) = 1; f (0) = 0; f (\frac{1}{2}) = \frac{1}{4}

Question 17

g (- 1000) = - 999; g (0) = 1; g (999) = 1000

Question 18

h (- \frac{12}{5}) = h (\frac{0}{1}) = h (\frac{9}{17}) = 2

Question 19

Since,

g (x) = \frac{2 x ^{3} + 2 x}{x ^{2} + 1} = \frac{2 x ( x ^{2} + 1 )}{x ^{2} + 1} = 2 x

Therefore for every $x \in R$ , $f (x) = g (x)$ , thus $f = g$ .

Question 20

Since,

K (x) = (x - 1) (x - 3) + 1 = x^{2} - 4 x + 4 = (x - 2)^{2}

Therefore for every $x \in R$ , $K (x) = H (x)$ , thus $K = H$ .

Exercise set 1.4

Question 1

$V (G) = {v_{1}, v_{2}, v_{3}, v_{4}}$
$E (G) = {e_{1}, e_{2}, e_{3}}$

$e_{1}$	${v_{1}, v_{2}}$
$e_{2}$	${v_{1}, v_{3}}$
$e_{3}$	${v_{3}}$

Question 2

$V (G) = {v_{1}, v_{2}, v_{3}, v_{4}}$
$E (G) = {e_{1}, e_{2}, e_{3}, e_{4}, e_{5}}$

$e_{1}$	${v_{1}, v_{2}}$
$e_{2}$	${v_{2}, v_{3}}$
$e_{3}$	${v_{2}, v_{3}}$
$e_{4}$	${v_{2}, v_{4}}$
$e_{5}$	${v_{4}}$

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

(i) $e_{1}, e_{2}, e_{3}$

(ii) $v_{1}, v_{2}$

(iii) $e_{2}, e_{3}, e_{8}, e_{9}$

(iv) $e_{6}, e_{7}$

(v) $e_{4}, e_{5}$ and $e_{8}, e_{9}$

(vi) $v_{6}$

(vii) $d e g (v_{3}) = 5$

Question 9

(i) $e_{1}, e_{2}, e_{7}$

(ii) $v_{1}, v_{2}$

(iii) $e_{2}, e_{7}$

(iv) $e_{1}, e_{3}$

(v) $e_{4}, e_{5}$

(vi) $v_{4}$

(vii) $d e g (v_{3}) = 2$

Question 10

(a) Yes.

(b) Yes.

Question 11

$(vv cc B /) \to (v c / B v c) \to (vv c B / c) \to (c / B vv c) \to (v c B / v c) \to (/ B vv cc)$
$(vv cc B /) \to (vv / B cc) \to (vv c B / c) \to (c / B vv c) \to (cc B / vv) \to (/ B vv cc)$
$(vv cc B /) \to (vv / B cc) \to (vv c B / c) \to (c / B vv c) \to (v c B / v c) \to (/ B vv cc)$

Question 12

$(w g c B /) \to (w c / B g) \to (w c B / g) \to$

$(c / Bw g) \to (c g B / w) \to$
$(w / B c g) \to (w g B / c) \to$

$(g / Bw c) \to (g B / w c) \to (/ B g w c)$

Question 13

$(ccc vvv B /) \to$

$(cc vv / B c v) \to$
$(vvv c / B cc) \to$

$(vvv cc B / c) \to (vvv / B ccc) \to (vvv c B / cc) \to (v c / B vv cc) \to (vv cc B / v c) \to$ $(cc / B vvv c) \to (ccc B / vvv) \to (c / B vvv cc) \to (cc B / vvv c) \to (/ B vvv ccc)$

Question 14

$(3/3, 0/5) \to$
- $(3/3, 5/5)$
- $(0/3, 3/5) \to (3/3, 3/5) \to (1/3, 5/5)$
$(0/3, 5/5) \to (3/3, 2/5)$

Question 15

Converting the map into a graph with vertices being the countries, the border between two countries is represented by an edge between them:

Question 16

Since the committees with the same color have no common member, we can plan the committees with the same color on the same time slots.

Question 17

Since the vertices (exams) of the same color have no common students (edge), they can be held in the same day.

Study Archive

Solutions 1 - Speaking Mathematically

Exercise set 1.1

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Exercise set 1.2

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Exercise set 1.3

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20

Exercise set 1.4

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17