(Quiz) Elementary Set Theory - Quick Review

Set theory is the basis of modern mathematics

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The German mathematician George Cantor (1845–1918) is regarded as the father of set theory

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The objects that belong to a set are called elements or members of the set

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From a mathematical point of view, a set is a well-defined collection of objects

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Valid vs Invalid set

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Mark each of the following sets as valid or invalid

(1) Set of natural numbers.
(2) Set of roots of the equation, x² + 7x + 6 = 0.
(3) A set consisting of prime minister of Australia, capital of the United States, natural numbers 1–10, Taj Mahal and alphabets a–c.
(4) Set of rich people.
(5) Set of cleaver students.

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valid

invalid

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The elements of a set need to be related to one another in an obvious way

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The sets are usually denoted by the small letters and its elements are denoted by the capital letters

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Element x belongs to set A

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If a particular element x belongs to set A, we write it as

x ∊ A

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Two elements x and y belong to set A

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If two elements x and y belong to set A, we shall write it as

x and y ∊ A

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Element x does not belong to set A

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If an element x does not belong to set A, we write it as

x ∉ A

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We use square brackets to enclose the elements of a set

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Expressing the set of all positive odd numbers

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The set of all positive odd numbers can written as

{ All positive odd numbers }
{ 1, 3, 5, 7, …. }
{ x|x is a positive odd number }

Format in point a is correct

Format in point b is correct

Format in point c is correct

Format in point b and c are correct

All formats in a, b, and c are correct

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What does symbol | mean in the following set

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A = { x|x is a positive even number }

therefore

such that

in other words

is explained as

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Set Notations

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Given the following set expressions, assign the right meaning to each symbol

x, y ∊ A
A = { x|x is an odd negative number } here -3 ∊ A, 8 ∉ A
A = ∅
N = {1, 2, 3, 4, ...}, n(N) = ℵ₀
R = {points in a plane}, n(R) = c
x ∊ A ⇔ x ∊ B

(1) capital letter A
(2) small letter x
(3) ∊
(4) ∉
(5) { } (curly brackets)
(6) | (pipe)
(7) , (comma) between x and y
(8) ∅
(9) ℵ₀(aleph-null)
(10) c
(11) ⇔

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enclosing elements of the set

denoting a set

denoting an element

cardinal number of countable infinite set

does not belong to

if and only if

The empty set

belongs to

and

cardinal number of an uncountable infinite set

such that

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A method used to represent a set in which all elements are listed within braces

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Roster Method

Rule Method

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A method used to represent a set in which we state one or more characteristic properties of the elements so that one is able to decide whether a given object is an element of the set

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Roster Method

Rule Method

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Singleton Set (Or Unit Set)

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A set that contains only one element is called __________

Missing

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Singleton set example

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Let A = { x|x + 5 = 5 }

set A = (1)

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{0}

{1}

{-1}

{}

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All null sets in different contexts are considered the same null set

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Singleton null set

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Assign the right term for each expression

A = ∅ (1)

A = {∅} (2)

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null set

singleton with null set

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The cardinal number of a set

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If a set A contains finite number of elements n, we denote the cardinal number of set A by n(A). In other words, n(A) stands for the number of elements in a finite set.

What are the cardinal number for each of the following sets?

A = {1, 3, 4, 5, 6, 7} (1)
B = {a, b, c, d, e, f, g, h} (2)
C = {2, 4, 6, 8, 10} (3)

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Cardinal number of countable and uncountable infinite set

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The cardinal number of (1) infinite set, N = {1, 2, 3, 4, ...} (or any other set that is equivalent to N), is denoted by the symbol ℵ₀ (read aleph-null). The symbol “c” is used to denote the cardinal number of an (2) infinite set, like the set of all real numbers or the set of points in an open interval or the set of points in a plane.

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countable

uncountable

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Cardinal number of the empty set

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What is the cardinal number of the empty set

n(∅) = (1)

Please type a number

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Subset of a set

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Given A = {1, 2, 3, 4, 5}, B = {1, 2, 3}, C = {2, 3, 1}, D = {6, 7, 8}. Mark true or false for the following

A ⊆ B (1)
B ⊆ A (2)
C ⊆ B (3)
B ⊆ C (4)
D ⊈ A (5)

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True

False

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What does the following symbolic form mean?

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B ⊇ A

subset

superset

belongs to

implies

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Set A is a subset of set B

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The set B is called the (1) of A

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Equality of sets

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x ∊ A ⇔ x ∊ B

The above statement means

Set A equals set B

Set A is subset of set B

Set B is a super set of A

Set A does not equals set B

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Every set has two subsets. The set itself and the null set

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Not a proper subset

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Consider A = {1, 2, 3} and B = {1, 2, 4, 5, 6}. Here, 3 ∈ A but 3 ∉ B. Hence, A ⊄ B.

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Two sets are said to be comparable if both are equal to each other

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Identify comparable and incomparable sets

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The sets A = {1, 2} and B = {1, 2, 3} are (1) as A ⊆ B. On the other hand, the sets C = {1, 2, 3} and D = {2, 3, 4} are (2).

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comparable

incomparable

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What do you call the below set?

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A = {∅, {1, 2}, {3}, {4, 5}}

set of elements

superset

set of sets

Invalid symbolic expression

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What is the power set of S?

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Let S = {1, 2, 3}, Then P(S) is

{{1}, {2}, {3}}

{{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}}

{{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

{∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

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What is the number of elements in the power set of S P(S)?

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S = {1, 2, 3}

3

6

8

10

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What is the number of elements in the power set of null set?

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∅ has (1) elements

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What is the set denoted by U?

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Universal Set

Universe of discourse

Union set

First and Second definitions are both correct

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Assign the right universal set to one of the following sets

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(1)	(2)
set of odd numbers	set of rational numbers
set of even numbers	set of irrational numbers
set of prime numbers
set of composite numbers
set of factors

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The set of natural numbers

The set of real numbers

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Arithmetic operations (e.g. addition, subtraction) are used in elementary set theory

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False

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The union of two sets

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A ∪ B = {x|x ∊ A (1) B (2) both}

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or

and

not in

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What is the union of two sets A and B?

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A = {1, 2, 3}, B = { 2, 3, 4, 5, 6}

{1, 2, 3}

{1, 2, 3, 4, 5, 6}

{4, 5, 6}

{2, 3, 4, 5, 6}

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Th union set A with empty set is A

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A ∪ ∅ = A

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If A is a subset of B, then A union B is B

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The intersection of two sets

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An example of an intersection operation

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Let A = {1, 2, 3}, B = {2, 3, 5, 6}, and C = {5, 7, 8}.

Complete the following intersection operations with the right values

A ∩ B = (1)
B ∩ C = (2)
A ∩ (3) = ∅

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{2,3}

{5}

{5, 6}

A

C

B

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No elements in common between sets

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Let A = {1, 2, 3} and B = {5, 6, 7}, then A ∩ B = ∅

The two sets are called 1($) sets

disconnected

non-related

disjoint

detached

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A - B reads as

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A minus B

A difference B

A excluding B

A not B

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A - B = B - A?

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If A = {1, 2, 3, 4}, B = {1, 3, 5, 6} THEN

A - B = B - A

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False

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A - B rule

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A – B = {x|x ∈ A, but x ∉ B}

A – B = {x|x ∉ A, but x ∈ B}

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Complement of a set

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Let U be the universal set and A is a proper subset of U, Then

A' = (1)

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A - U

U - A

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Complement of a set - special cases

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Let U the universal set, A a proper subset of U

(A')^' = U - A' = U - ((1)) = A
∅' = (2) - ∅ = (3)
U' = (4) = ∅

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U - A

U

U - U

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Complement of a set - example

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Let N = {1, 2, 3, 4, ...} = U and A' = {2, 4, 6, 8, ...}, then A =

{1, 3, 5, 7, ...}

{0, 1, 3, 5, 7, ...}

{1, 2, 3, 4, 5, 6, ...}

{0, 2, 4, 6, ...}

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