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Quiz 1: Numbers and sets

Question

Question 1

Which of the following are correct ways of writing the set

A = {x \in ℝ | - 3 < x \leq - 1 o r x \geq 0} ?

Not correct. Choice (a) is false.

(- 3, \infty)

includes the real numbers between

- 1

and

0

,
which do not belong to

A

Not correct. Choice (b) is false.

[- 3, \infty)

includes

- 3

, as well as the real numbers between

- 1

and

0

, none of which belong to

A

Not correct. Choice (c) is false.

[- 3, - 1] \cap [0, \infty)

is the empty set

\emptyset

. As

A

is not empty (for example,

A

includes

- 1

), this option can’t be correct.

Not correct. Choice (d) is false.

- 1

is not in

(- 3, - 1) \cup [0, \infty)

, but

- 1

is in

A

Your answer is correct.

Question 2

What is another way of writing the set

B = {x \in ℝ | | x - 3 | < 2} ?

Not correct. Choice (a) is false.

For example,

4

belongs to

B

but is not in

(2, 3]

Not correct. Choice (b) is false.

For example,

1.5

belongs to

B

but is not in

[2, 4]

Your answer is correct.

B

is the set of all points whose distance from 3 on the number line is less than 2.
The solution to

| x - 3 | < 2

1 < x < 5

Not correct. Choice (d) is false.

Neither

1

nor

5

belong to

B

, but both

1

and

5

belong to

[1, 5]

Not correct. Choice (e) is false.

For example,

4

belongs to

B

but is not in

[2, 3)

Question 3

A = {7, 8, 9, 10}

and

B = {5, 6, 7, 8}

then

(A \ B) \cup (B \ A)

Not correct. Choice (a) is false.

Your answer is correct.

A \ B = {9, 10}

and

B \ A = {5, 6}

(A \ B) \cup (B \ A) = {5, 6, 9, 10}

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Not correct. Choice (e) is false.

Question 4

The set

{0, 1, \pm \sqrt{- 1}, π, 12}

is a subset of

Not correct. Choice (a) is false.

The number

π

is not a natural number.

Not correct. Choice (b) is false.

The number

π

is not an integer.

Not correct. Choice (c) is false.

The number

π

is not a rational number.

Not correct. Choice (d) is false.

\sqrt{- 1}

is not real.

Your answer is correct.

Since

\pm \sqrt{- 1}

denotes the two imaginary numbers

i

and

- i

, the given set cannot be in any of the sets

ℕ, ℤ, ℚ o r ℝ

.
Hence the right answer must be

ℂ

which contains all imaginary numbers.

Question 5

Which of the following alternatives is the best response to ‘Solve

x^{2} - 3 x + 4 = 0

over

ℂ

’.

Not correct. Choice (a) is false.

As the question asks us to solve the equation over

ℂ

(that is, to find all solutions belonging to the set of complex numbers), this is not the best response.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Your answer is correct.

Using the quadratic formula,

x = \frac{3 \pm \sqrt{9 - 16}}{2} = \frac{3 \pm \sqrt{- 7}}{2}

Not correct. Choice (e) is false.

Question 6

z = 9 + 3 i

and

w = 2 - i

then

z + w

equals

Not correct. Choice (a) is false.

Your answer is correct.

z + w = (9 + 3 i) + (2 - i) = (9 + 2) + (3 - 1) i = 11 + 2 i

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Not correct. Choice (e) is false.

Question 7

w = 2 - i

then

\bar{w}

equals

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Your answer is correct.

\bar{w} = \bar{2 - i} = 2 + i

Not correct. Choice (d) is false.

Not correct. Choice (e) is false.

Question 8

p = 9 + 3 i

and

q = 2 - i

then

p \bar{q}

equals

Your answer is correct.

p \bar{q} = (9 + 3 i) \bar{(2 - i)}

= (9 + 3 i) (2 + i) = (18 - 3) + (6 + 9) i

= 15 + 15 i

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Not correct. Choice (e) is false.

Question 9

z = 9 + 3 i

and

w = 2 - i

then

\frac{z}{w}

equals

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Your answer is correct.

\frac{z}{w} = \frac{9 + 3 i}{2 - i} = \frac{9 + 3 i}{2 - i} \times \frac{2 + i}{2 + i} = \frac{15 + 15 i}{5} = 3 + 3 i

Question 10

The shaded region in the graph

corresponds to which set of complex numbers?

Your answer is correct.

Not correct. Choice (b) is false.

This set corresponds to the interior of a circle, centre the origin, radius

2 + \sqrt{2}

Not correct. Choice (c) is false.

This set corresponds to the open half plane containing all complex numbers

z = x + i y

with

x < 1

Not correct. Choice (d) is false.

This set corresponds to the interior of a circle, centre

2

, radius

\sqrt{2}

Not correct. Choice (e) is false.

ABN: 15 211 513 464. CRICOS number: 00026A. Phone: +61 2 9351 2222.

Authorised by: Head, School of Mathematics and Statistics.

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Questions:

right first attempt
right
wrong

a)	$(- 3, \infty)$	b)	$[- 3, \infty)$
c)	$[- 3, - 1] \cap [0, \infty)$	d)	$(- 3, - 1) \cup [0, \infty)$
e)	$(- 3, - 1] \cup [0, \infty)$

a)	${5, 6, 7, 8, 9, 10}$	b)	${5, 6, 9, 10}$
c)	$\emptyset$ , the empty set.	d)	${7, 8}$
e)	None of the above.

a)	There are no real solutions.	b)	$x = 1, 4$
c)	$x = \frac{3 \pm \sqrt{7}}{2}$	d)	$x = \frac{3 \pm i \sqrt{7}}{2}$
e)	None of the above is correct.

a)	$15 + 15 i$	b)	$21 + 15 i$
c)	$18 + 3 i$	d)	$1 - i$
e)	None of the above

a)	$15 + 15 i$	b)	$6 + 3 i$
c)	$12 + 15 i$	d)	$3 - 3 i$
e)	None of the above

a)	${z \in ℂ : \| z - (i + 1) \| < 2}$	b)	${z \in ℂ : \| z \| - \| 1 + i \| < 2}$
c)	${z \in ℂ : Re (z + (i + 1)) < 2}$	d)	${z \in ℂ : \| z - 2 \| < \| i + 1 - 2 \|}$
e)	None of the above.

a)	$(2, 3]$	b)	$[2, 4]$
c)	$(1, 5)$	d)	$[1, 5]$
e)	$[2, 3)$

a)	$ℕ$	b)	$ℤ$
c)	$ℚ$	d)	$ℝ$
e)	$ℂ$

The University of Sydney - School of Mathematics and Statistics

Quiz 1: Numbers and sets

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

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