WGU C959 DISCRETE MATHEMATICS
COMPREHENSIVE EXAMINATION TEST WITH
COMPLETE QUESTIONS AND SOLUTIONS
◉ Can an element of a set itself be a set? Answer: Yes, a set can
contain elements that are themselves sets. (Section 2.2)
◉ When counting the cardinality of a set whose elements include
other sets, how is each inner set counted? Answer: Each set contained
within the outer set counts as a single element, regardless of how
many elements that inner set itself contains. (Section 2.2)
◉ Is the empty set the same thing as a set containing only the empty
set as its single element? Answer: No, the empty set has a cardinality
of 0, while a set containing only the empty set as its one element has a
cardinality of 1. (Section 2.2)
◉ What is the difference between using the "is an element of" symbol
(∈) versus the "is a subset of" symbol (⊆) when relating a set to
another set of sets? Answer: The element symbol is used when the set
in question is literally one of the listed members of the outer set,
while the subset symbol is used when every member of the set in
question is also a member of the outer set. (Section 2.2)
,◉ Can a set contain a mix of individual numbers and sets of numbers
as its elements? Answer: Yes, a single set can contain a combination
of plain numbers and other sets as its elements at the same time.
(Section 2.2)
◉ What is the power set of a set A? Answer: The set of all subsets of
A, including the empty set and A itself. (Section 2.2)
◉ What notation is commonly used to denote the power set of a set
A? Answer: P(A). (Section 2.2)
◉ According to the theorem on the cardinality of a power set, if a
finite set A has cardinality n, what is the cardinality of its power set?
Answer: 2 raised to the power of n. (Section 2.2)
◉ Why does the empty set always appear as an element of the power
set of any set? Answer: Because the empty set is considered a subset
of every set, so it must be included among the subsets that make up
the power set. (Section 2.2)
◉ Why does a set A always appear as an element of its own power
set, P(A)? Answer: Because a set is always considered a subset of
itself, so A itself must be included among the subsets that make up the
power set. (Section 2.2)
◉ If a set has 4 elements, how many total subsets does it have,
including the empty set and the full set itself? Answer: 16 subsets,
since 2 raised to the power of 4 equals 16. (Section 2.2)
,◉ When listing out the elements of a power set of a set with several
elements, why is it helpful to organize the subsets by size? Answer:
Organizing subsets by size, such as listing all subsets of size 0, then
size 1, then size 2, and so on, makes it easier to systematically ensure
that every possible subset has been accounted for without missing or
repeating any. (Section 2.2)
◉ What is the intersection of two sets A and B? Answer: The set of all
elements that are elements of both A and B. (Section 2.3)
◉ How is the intersection of sets A and B denoted, and how is it read
aloud? Answer: It is denoted A ∩ B, read as "A intersect B." (Section
2.3)
◉ What is the union of two sets A and B? Answer: The set of all
elements that are elements of A or B. (Section 2.3)
◉ How is the union of sets A and B denoted, and how is it read aloud?
Answer: It is denoted A ∪ B, read as "A union B." (Section 2.3)
◉ Does the definition of union use the inclusive or or the exclusive
or? Answer: The inclusive or, meaning that if an element belongs to
both A and B, it is still included as a single element in the union.
(Section 2.3)
, ◉ Can the intersection and union operations be applied to infinite
sets, not just finite ones? Answer: Yes, both the intersection and union
operations can be applied to infinite sets, such as sets defined by
properties like being multiples of a certain number. (Section 2.3)
◉ Why is it important to use parentheses when combining more than
one set operation, such as in an expression involving both union and
intersection? Answer: Because the order in which different types of
operations, like union and intersection, are applied can change the
resulting set, so parentheses are needed to make the intended order of
operations clear. (Section 2.3)
◉ When evaluating an expression like A ∩ (B ∪ C), which operation
should be performed first? Answer: The operation inside the
parentheses, the union of B and C, should be performed first, and then
the intersection with A is taken of that result. (Section 2.3)
◉ Why is an expression involving multiple intersection operations on
several sets, such as A ∩ B ∩ C ∩ D, considered well-defined without
needing parentheses? Answer: Because the order in which multiple
intersection operations are applied does not affect the final resulting
set. (Section 2.3)
◉ Why is an expression involving multiple union operations on
several sets, such as A ∪ B ∪ C ∪ D, considered well-defined without
needing parentheses? Answer: Because the order in which multiple
union operations are applied does not affect the final resulting set.
(Section 2.3)
COMPREHENSIVE EXAMINATION TEST WITH
COMPLETE QUESTIONS AND SOLUTIONS
◉ Can an element of a set itself be a set? Answer: Yes, a set can
contain elements that are themselves sets. (Section 2.2)
◉ When counting the cardinality of a set whose elements include
other sets, how is each inner set counted? Answer: Each set contained
within the outer set counts as a single element, regardless of how
many elements that inner set itself contains. (Section 2.2)
◉ Is the empty set the same thing as a set containing only the empty
set as its single element? Answer: No, the empty set has a cardinality
of 0, while a set containing only the empty set as its one element has a
cardinality of 1. (Section 2.2)
◉ What is the difference between using the "is an element of" symbol
(∈) versus the "is a subset of" symbol (⊆) when relating a set to
another set of sets? Answer: The element symbol is used when the set
in question is literally one of the listed members of the outer set,
while the subset symbol is used when every member of the set in
question is also a member of the outer set. (Section 2.2)
,◉ Can a set contain a mix of individual numbers and sets of numbers
as its elements? Answer: Yes, a single set can contain a combination
of plain numbers and other sets as its elements at the same time.
(Section 2.2)
◉ What is the power set of a set A? Answer: The set of all subsets of
A, including the empty set and A itself. (Section 2.2)
◉ What notation is commonly used to denote the power set of a set
A? Answer: P(A). (Section 2.2)
◉ According to the theorem on the cardinality of a power set, if a
finite set A has cardinality n, what is the cardinality of its power set?
Answer: 2 raised to the power of n. (Section 2.2)
◉ Why does the empty set always appear as an element of the power
set of any set? Answer: Because the empty set is considered a subset
of every set, so it must be included among the subsets that make up
the power set. (Section 2.2)
◉ Why does a set A always appear as an element of its own power
set, P(A)? Answer: Because a set is always considered a subset of
itself, so A itself must be included among the subsets that make up the
power set. (Section 2.2)
◉ If a set has 4 elements, how many total subsets does it have,
including the empty set and the full set itself? Answer: 16 subsets,
since 2 raised to the power of 4 equals 16. (Section 2.2)
,◉ When listing out the elements of a power set of a set with several
elements, why is it helpful to organize the subsets by size? Answer:
Organizing subsets by size, such as listing all subsets of size 0, then
size 1, then size 2, and so on, makes it easier to systematically ensure
that every possible subset has been accounted for without missing or
repeating any. (Section 2.2)
◉ What is the intersection of two sets A and B? Answer: The set of all
elements that are elements of both A and B. (Section 2.3)
◉ How is the intersection of sets A and B denoted, and how is it read
aloud? Answer: It is denoted A ∩ B, read as "A intersect B." (Section
2.3)
◉ What is the union of two sets A and B? Answer: The set of all
elements that are elements of A or B. (Section 2.3)
◉ How is the union of sets A and B denoted, and how is it read aloud?
Answer: It is denoted A ∪ B, read as "A union B." (Section 2.3)
◉ Does the definition of union use the inclusive or or the exclusive
or? Answer: The inclusive or, meaning that if an element belongs to
both A and B, it is still included as a single element in the union.
(Section 2.3)
, ◉ Can the intersection and union operations be applied to infinite
sets, not just finite ones? Answer: Yes, both the intersection and union
operations can be applied to infinite sets, such as sets defined by
properties like being multiples of a certain number. (Section 2.3)
◉ Why is it important to use parentheses when combining more than
one set operation, such as in an expression involving both union and
intersection? Answer: Because the order in which different types of
operations, like union and intersection, are applied can change the
resulting set, so parentheses are needed to make the intended order of
operations clear. (Section 2.3)
◉ When evaluating an expression like A ∩ (B ∪ C), which operation
should be performed first? Answer: The operation inside the
parentheses, the union of B and C, should be performed first, and then
the intersection with A is taken of that result. (Section 2.3)
◉ Why is an expression involving multiple intersection operations on
several sets, such as A ∩ B ∩ C ∩ D, considered well-defined without
needing parentheses? Answer: Because the order in which multiple
intersection operations are applied does not affect the final resulting
set. (Section 2.3)
◉ Why is an expression involving multiple union operations on
several sets, such as A ∪ B ∪ C ∪ D, considered well-defined without
needing parentheses? Answer: Because the order in which multiple
union operations are applied does not affect the final resulting set.
(Section 2.3)
