MTH 101 Flashcards
What is a set in mathematics?
A set is a collection of distinct objects, considered as an object in its own right.
True or False: A set can contain duplicate elements.
False
What symbol is commonly used to denote a set?
{}
Fill in the blank: The elements of a set are called _____.
members
What is the empty set?
The empty set is a set that contains no elements, denoted by {} or ∅.
What is the union of two sets?
The union of two sets is the set containing all elements from both sets.
If A = {1, 2, 3} and B = {3, 4, 5}, what is A ∪ B?
{1, 2, 3, 4, 5}
What is the intersection of two sets?
The intersection of two sets is the set containing all elements that are common to both sets.
If A = {1, 2, 3} and B = {2, 3, 4}, what is A ∩ B?
{2, 3}
What does it mean for a set to be a subset?
A set A is a subset of set B if every element of A is also an element of B.
True or False: Every set is a subset of itself.
True
What is the power set of a set?
The power set of a set is the set of all possible subsets of that set.
If A = {1, 2}, what is the power set of A?
{{}, {1}, {2}, {1, 2}}
What is the complement of a set?
The complement of a set A refers to elements not in A but in a universal set U.
If U = {1, 2, 3, 4, 5} and A = {1, 2}, what is the complement of A?
{3, 4, 5}
What is a universal set?
A universal set is the set that contains all possible elements for a particular discussion.
Define disjoint sets.
Disjoint sets are sets that have no elements in common.
If A = {1, 2} and B = {3, 4}, are A and B disjoint?
Yes
What is a finite set?
A finite set is a set with a countable number of elements.
What is an infinite set?
An infinite set is a set with an uncountable number of elements.
Give an example of an infinite set.
The set of all natural numbers.
What are equivalent sets?
Equivalent sets are sets that have the same number of elements.
True or False: Two sets can be equivalent even if they contain different elements.
True
What does it mean for two sets to be equal?
Two sets are equal if they contain exactly the same elements.
If A = {1, 2} and B = {2, 1}, are A and B equal?
Yes
What notation is used to denote a set of all x such that x satisfies a certain property?
Set-builder notation.
Provide an example of a set defined in set-builder notation.
{x | x is an even integer}
What is a Cartesian product of two sets?
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.
If A = {1, 2} and B = {x, y}, what is A × B?
{(1, x), (1, y), (2, x), (2, y)}
What is the difference between two sets?
The difference between two sets A and B is the set of elements that are in A but not in B.
If A = {1, 2, 3} and B = {2, 3, 4}, what is A - B?
{1}
What is a multi-set?
A multi-set is a generalization of a set that allows for multiple occurrences of the same element.
True or False: In a multi-set, the order of elements matters.
False
What is a Venn diagram?
A Venn diagram is a diagram that shows all possible logical relations between a finite collection of sets.
What does the symbol ∈ represent?
The symbol ∈ represents ‘is an element of’.
What does the symbol ∉ represent?
The symbol ∉ represents ‘is not an element of’.
If A = {a, b, c}, is b ∈ A?
Yes
If A = {a, b, c}, is d ∉ A?
Yes
What is a finite union of sets?
A finite union of sets is the union operation applied to a finite number of sets.
What is a finite intersection of sets?
A finite intersection of sets is the intersection operation applied to a finite number of sets.
Describe the symmetric difference of two sets.
The symmetric difference of two sets A and B is the set of elements in either A or B but not in both.
If A = {1, 2} and B = {2, 3}, what is A Δ B?
{1, 3}
What is a countable set?
A countable set is a set that can be put into a one-to-one correspondence with the natural numbers.
What is an uncountable set?
An uncountable set is a set that cannot be put into a one-to-one correspondence with the natural numbers.
Give an example of a countable set.
The set of integers.
Give an example of an uncountable set.
The set of real numbers.
What is a set operation?
A set operation is an operation that combines or relates sets, such as union, intersection, or difference.
What is a relation in set theory?
A relation is a set of ordered pairs of elements from two sets.
What is a function in set theory?
A function is a specific type of relation where each element in the domain is related to exactly one element in the codomain.
What does it mean for a function to be injective?
A function is injective if it maps distinct elements of the domain to distinct elements of the codomain.
What does it mean for a function to be surjective?
A function is surjective if every element of the codomain is mapped to by at least one element of the domain.
What does it mean for a function to be bijective?
A function is bijective if it is both injective and surjective.
What is a sequence in set theory?
A sequence is an ordered list of elements, where elements can repeat.
What is a tuple?
A tuple is an ordered collection of elements, which can be of different types.
What is the difference between a set and a list?
A set is unordered and contains unique elements, while a list is ordered and can contain duplicates.
What is a finite sequence?
A finite sequence is a sequence that has a definite number of terms.
What is an infinite sequence?
An infinite sequence is a sequence that continues indefinitely.
What is a series in mathematics?
A series is the sum of the terms of a sequence.
What is a convergent series?
A convergent series is a series whose terms approach a specific value as the number of terms increases.
What is a divergent series?
A divergent series is a series that does not converge to a specific value.
What is a cardinality of a set?
The cardinality of a set is the number of elements in the set.
If A = {1, 2, 3}, what is the cardinality of A?
3
What is the difference between finite and infinite cardinality?
Finite cardinality is a countable number of elements, while infinite cardinality represents an uncountable number of elements.
What is a countably infinite set?
A countably infinite set is an infinite set that can be put into one-to-one correspondence with the natural numbers.
What is a Cantor set?
A Cantor set is a type of set that is uncountable and has no intervals, created by repeatedly removing the middle third of intervals.
