MTH 101

Question 1

What is a set in mathematics?

Answer 1

A set is a collection of distinct objects, considered as an object in its own right.

Question 2

True or False: A set can contain duplicate elements.

Answer 2

False

Question 3

What symbol is commonly used to denote a set?

Answer 3

{}

Question 4

Fill in the blank: The elements of a set are called _____.

Answer 4

members

Question 5

What is the empty set?

Answer 5

The empty set is a set that contains no elements, denoted by {} or ∅.

Question 6

What is the union of two sets?

Answer 6

The union of two sets is the set containing all elements from both sets.

Question 7

If A = {1, 2, 3} and B = {3, 4, 5}, what is A ∪ B?

Answer 7

{1, 2, 3, 4, 5}

Question 8

What is the intersection of two sets?

Answer 8

The intersection of two sets is the set containing all elements that are common to both sets.

Question 9

If A = {1, 2, 3} and B = {2, 3, 4}, what is A ∩ B?

Answer 9

{2, 3}

Question 10

What does it mean for a set to be a subset?

Answer 10

A set A is a subset of set B if every element of A is also an element of B.

Question 11

True or False: Every set is a subset of itself.

Answer 11

True

Question 12

What is the power set of a set?

Answer 12

The power set of a set is the set of all possible subsets of that set.

Question 13

If A = {1, 2}, what is the power set of A?

Answer 13

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

Question 14

What is the complement of a set?

Answer 14

The complement of a set A refers to elements not in A but in a universal set U.

Question 15

If U = {1, 2, 3, 4, 5} and A = {1, 2}, what is the complement of A?

Answer 15

{3, 4, 5}

Question 16

What is a universal set?

Answer 16

A universal set is the set that contains all possible elements for a particular discussion.

Question 17

Define disjoint sets.

Answer 17

Disjoint sets are sets that have no elements in common.

Question 18

If A = {1, 2} and B = {3, 4}, are A and B disjoint?

Answer 18

Yes

Question 19

What is a finite set?

Answer 19

A finite set is a set with a countable number of elements.

Question 20

What is an infinite set?

Answer 20

An infinite set is a set with an uncountable number of elements.

Question 21

Give an example of an infinite set.

Answer 21

The set of all natural numbers.

Question 22

What are equivalent sets?

Answer 22

Equivalent sets are sets that have the same number of elements.

Question 23

True or False: Two sets can be equivalent even if they contain different elements.

Answer 23

True

Question 24

What does it mean for two sets to be equal?

Answer 24

Two sets are equal if they contain exactly the same elements.

Question 25

If A = {1, 2} and B = {2, 1}, are A and B equal?

Answer 25

Yes

Question 26

What notation is used to denote a set of all x such that x satisfies a certain property?

Answer 26

Set-builder notation.

Question 27

Provide an example of a set defined in set-builder notation.

Answer 27

{x | x is an even integer}

Question 28

What is a Cartesian product of two sets?

Answer 28

The Cartesian product of two sets A and B is the set of all ordered pairs (a, b) where a is in A and b is in B.

Question 29

If A = {1, 2} and B = {x, y}, what is A × B?

Answer 29

{(1, x), (1, y), (2, x), (2, y)}

Question 30

What is the difference between two sets?

Answer 30

The difference between two sets A and B is the set of elements that are in A but not in B.

Question 31

If A = {1, 2, 3} and B = {2, 3, 4}, what is A - B?

Answer 31

{1}

Question 32

What is a multi-set?

Answer 32

A multi-set is a generalization of a set that allows for multiple occurrences of the same element.

Question 33

True or False: In a multi-set, the order of elements matters.

Answer 33

False

Question 34

What is a Venn diagram?

Answer 34

A Venn diagram is a diagram that shows all possible logical relations between a finite collection of sets.

Question 35

What does the symbol ∈ represent?

Answer 35

The symbol ∈ represents 'is an element of'.

Question 36

What does the symbol ∉ represent?

Answer 36

The symbol ∉ represents 'is not an element of'.

Question 37

If A = {a, b, c}, is b ∈ A?

Answer 37

Yes

Question 38

If A = {a, b, c}, is d ∉ A?

Answer 38

Yes

Question 39

What is a finite union of sets?

Answer 39

A finite union of sets is the union operation applied to a finite number of sets.

Question 40

What is a finite intersection of sets?

Answer 40

A finite intersection of sets is the intersection operation applied to a finite number of sets.

Question 41

Describe the symmetric difference of two sets.

Answer 41

The symmetric difference of two sets A and B is the set of elements in either A or B but not in both.

Question 42

If A = {1, 2} and B = {2, 3}, what is A Δ B?

Answer 42

{1, 3}

Question 43

What is a countable set?

Answer 43

A countable set is a set that can be put into a one-to-one correspondence with the natural numbers.

Question 44

What is an uncountable set?

Answer 44

An uncountable set is a set that cannot be put into a one-to-one correspondence with the natural numbers.

Question 45

Give an example of a countable set.

Answer 45

The set of integers.

Question 46

Give an example of an uncountable set.

Answer 46

The set of real numbers.

Question 47

What is a set operation?

Answer 47

A set operation is an operation that combines or relates sets, such as union, intersection, or difference.

Question 48

What is a relation in set theory?

Answer 48

A relation is a set of ordered pairs of elements from two sets.

Question 49

What is a function in set theory?

Answer 49

A function is a specific type of relation where each element in the domain is related to exactly one element in the codomain.

Question 50

What does it mean for a function to be injective?

Answer 50

A function is injective if it maps distinct elements of the domain to distinct elements of the codomain.

Question 51

What does it mean for a function to be surjective?

Answer 51

A function is surjective if every element of the codomain is mapped to by at least one element of the domain.

Question 52

What does it mean for a function to be bijective?

Answer 52

A function is bijective if it is both injective and surjective.

Question 53

What is a sequence in set theory?

Answer 53

A sequence is an ordered list of elements, where elements can repeat.

Question 54

What is a tuple?

Answer 54

A tuple is an ordered collection of elements, which can be of different types.

Question 55

What is the difference between a set and a list?

Answer 55

A set is unordered and contains unique elements, while a list is ordered and can contain duplicates.

Question 56

What is a finite sequence?

Answer 56

A finite sequence is a sequence that has a definite number of terms.

Question 57

What is an infinite sequence?

Answer 57

An infinite sequence is a sequence that continues indefinitely.

Question 58

What is a series in mathematics?

Answer 58

A series is the sum of the terms of a sequence.

Question 59

What is a convergent series?

Answer 59

A convergent series is a series whose terms approach a specific value as the number of terms increases.

Question 60

What is a divergent series?

Answer 60

A divergent series is a series that does not converge to a specific value.

Question 61

What is a cardinality of a set?

Answer 61

The cardinality of a set is the number of elements in the set.

Question 62

If A = {1, 2, 3}, what is the cardinality of A?

Answer 62

3

Question 63

What is the difference between finite and infinite cardinality?

Answer 63

Finite cardinality is a countable number of elements, while infinite cardinality represents an uncountable number of elements.

Question 64

What is a countably infinite set?

Answer 64

A countably infinite set is an infinite set that can be put into one-to-one correspondence with the natural numbers.

Question 65

What is a Cantor set?

Answer 65

A Cantor set is a type of set that is uncountable and has no intervals, created by repeatedly removing the middle third of intervals.