Topic 1.2 Mathematical Sets

Question 1

What information should come to mind when faced with this set (N)?

Answer 1

That symbol denotes the set of natural numbers/non-standard integers e.g. {1, 2, 3, 4…}

Question 2

What information should come to mind when faced with this set (Z)?

Answer 2

That symbol denotes the set of integers, which consist of the positive and negative whole numbers as well as zero e.g. {… -3, -2, -1, 0, 1, 2, 3…}

Question 3

What information should come to mind when faced with this set (Q)?

Answer 3

That symbol denotes the set of rational numbers (recall, every integer is a rational number, as it can be expressed as a fraction with denominator = 1). This set consists of numbers that can be written as fractions m/n where m and n are integers e.g. 1/2, 78/23, (−3)/4, 0/5

Question 4

What information should come to mind when faced with this set (R)?

Answer 4

That symbol denotes the set of real numbers (although, that could lead one to question what counts as a real number at all). The set consists of all the rational numbers together along with all the irrational numbers.

Question 5

Provide a definition for real numbers

Answer 5

Any limits of sequences of real numbers are real numbers

Question 6

How would the following statement read? x∈A

Answer 6

‘x’ is in set A

Question 7

How would the following statement read? π∉Q

Answer 7

‘π’ is not in set Q

Question 8

In what ways do sets vary from lists?

Answer 8

They vary in terms of order and repetition i.e. order/repetition doesn’t matter in sets; order/repetition matters in lists

Question 9

How would you represent a tuple?

Answer 9

With rounded brackets e.g. (a, b, c), where (a, b, c) ≠ (a, c, b) ≠ (a, a, b, c)

Question 10

Why is the following reasonable? {0, 0, 0, 1, 2, 7, 9} = {0, 1, 2, 7, 9}

Answer 10

Sets contain unique objects, which is why repetition and order don’t matter

Question 11

Why are the following incorrect? 2⊆Z {2}∈R

Answer 11

Because these statements imply that there are subsets in sets, sets that have specified no such thing

Question 12

If A and B are sets, when would A = B

Answer 12

When A⊆B & B⊆A

Question 13

How would one read the following statement? S={x∈R|sin⁡(x)>0}

Answer 13

The line in the middle is read as ‘such that’. According to the statement, the set should only contain elements of R that obeyed the condition sin⁡(x)>0

Question 14

What’s the difference between open and closed brackets?

Answer 14

Closed brackets [] means the element it encloses is put into consideration, whereas open brackets () means the element it encloses isn’t put into consideration e.g. a is included but b isn’t [a, b)

Question 15

Let A be the set containing all integer multiples of π. Express A using set builder notation

Answer 15

A={x∈R|∃k∈Z x=kπ}

Question 16

Let A be the set containing all integer multiples of π. Express A using abbreviated set builder notation

Answer 16

A={kπ|k∈Z}

Question 17

What is the abbreviated set builder notation for the set of rational numbers?

Answer 17

Q={m/n|m,n∈Z, n≠0}

Question 18

How do you express when all elements of set A are also elements of set B and vice versa?

Answer 18

A = B

Question 19

How do you know when A is a subset of B?

Answer 19

When every element of A is an element of B

Question 20

How do you express that A is a subset of B? What does this mean?

Answer 20

Expressed as A⊆B
This means that if x⊆A, then x⊆B (∀x⊆A x⊆B)

Question 21

Is the following true? Q∈R

Answer 21

No. Rational numbers are a subset of real numbers, they are not ‘in’ real numbers i.e. Q⊆R

Question 22

Assume A and B are two distinct sets. What does A∪B mean?

Answer 22

A∪B means the union of sets A and B, whereby the union is a set of elements that are in at least one of A or B (x∈A∪B if and only if x∈A or x∈B)

Question 23

Assume A and B are two distinct sets. What does A∩B mean?

Answer 23

A∩B means the intersection of sets A and B, forming a set comprised of the elements that are in both A and B (x∈A∩B if and only if x∈A and x∈B aka A∩B={x∈A│x∈B}={x∈B│x∈A})

Question 24

What is the notation for empty sets? Provide an example

Answer 24

{} = ∅
(−1,2)∩(100,101)=∅

Question 25

Let A and B be two distinct sets. What is the Cartesian product of A and B and how would you denote it?

Answer 25

The Cartesian product is the set of all possble ordered pairs we can build using elements from a defined set, in this case, we would use elements of A as the first element in the ordered pair, and we would use elements of B as the second element in the pair.
A ×B={(a,b)|a∈A,b∈B}