set theory

Question 1

by convention, how do you denote a set?

Answer 1

using capital letters

Question 2

3 ways to define a set

Answer 2

1. describe the set in words
2. list the elements of the set in curly brackets (such that a pattern is clear)
3. describe the elements of the set (typically using a small letter) using the set-builder notation

Question 3

what type of number is zero?

Answer 3

neutral

neither positive nor negative

Question 4

what do ∈ and ∉ mean respectively?

∈ is a variant of the actual epsilon letter.

Answer 4

∈: “is an element of/belongs to”
∉: “is not an element of/does not belong to”

Question 5

what does N represent?

Answer 5

set of natural/counting numbers
e.g. 1, 2, 3, 4…

Question 6

what do Z, Z+ and Z- represent?

Answer 6

• Z is the set of all integers.
• Z+ is the set of all positive integers (1, 2, 3, …)
• Z- is the set of all negative integers (…, -3, -2, -1).
• Zero is not included in either of these sets .
• Znonneg is the set of all positive integers including 0
• Znonpos is the set of all negative integers including 0.

Question 7

what does Q represent?

Answer 7

set of all rational numbers
(number that can be expressed as the quotient/fraction ⁠⁠of two integers, a numerator p and a non-zero denominator q)

Question 8

what does R represent?

Answer 8

set of real numbers

Question 9

what does … denote?

Answer 9

…: numbers continue in the same pattern infinitely

can be put on both sides of a set

Question 10

define:

equal set

Answer 10

two sets are equal if they contain exactly the same elements

Question 10

define

disjoint sets

Answer 10

sets with no common elements

Question 10

define:

empty/null set

Answer 10

set with no elements, denoted by ∅

Question 11

define:

finite and infinite set

Answer 11

• finite set: can list all the elements in the set. n(X), n represents the no. of elements.
• infinite set: cannot list all the elements in the set/infinitely many elements. e.g. Z, the set of integers

Question 12

define

subsets & proper subsets

Answer 12

subsets: every element in B is also an element of A, denoted by ⊆

proper subsets: subset + n(B) ≤ n(A)/there are elements in A not found in B, denoted by ⊂

when A is a subset of B, then n(A) less or equal than n(B). (so the number of elements of A could be 0!)

∅ is a subset of any set.

Question 13

define

universal set & complement of a set

Answer 13

universal set: contains all the elements in a discussion
complement of a set: within the universal set, say there is a set A. the complement of a set contains all the elements in the universal set that are not elements of A