WGU C960 DISCRETE MATH II OA QUESTIONS
AND SOLUTIONS GRADED A+
◉ proofs by contrapositive
Answer: proves a conditional theorem of the form p → q by showing
that the contrapositive ¬q → ¬p is true. In other words, ¬c is
assumed to be true and ¬p is proven as a result of ¬q.
Logically equivalent to if p then q
◉ proof by contradiction
Answer: (indirect proof)
starts by assuming that the theorem is false and then shows that
some logical inconsistency arises as a result of this assumption.
Notice not a conditional.
Want to prove Y
Assume not Y
Find a contradiction in X and Not Y
Therefore, claim not not Y.
◉ proof by cases
Answer: A proof by cases of a universal statement such as ∀x P(x)
breaks the domain for the variable x into different classes and gives
,a different proof for each class. Every value in the domain must be
included in at least one class.
◉ Unit 2 sets and functions
Answer:
◉ object in a set are called
Answer: elements
◉ The symbol ∈ is used:
Answer: to indicate that an element is in a set, as in 2 ∈ A
◉ The set with no elements is called the empty set and is denoted by
the symbol ∅.
Answer: The empty set is sometimes referred to as the null setand
can also be denoted by {}. Because the empty set has no elements,
for any element a, a ∉ ∅ is true.
◉ the cardinality of the empty set is:
Answer: zero
◉N
,Answer: The set of natural numbers: All integers greater than or
equal to 0. 0, 1, 2, ...
◉Z
Answer: The set of all integers. ..., -2, -1, 0, 1, 2, ...
◉Q
Answer: The set of rational numbers: All real numbers that can be
expressed as a/b, where a and b are integers and b ≠ 0. 0, 1/2, 5.23, -
5/3
◉R
Answer: The set of real numbers.
◉ The superscript + is used to indicate the positive elements of a
particular set.
Answer: For example, the set R+ is the set of all positive real
numbers, and Z+ is the set of all positive integers.
◉ The universal set, usually denoted by the variable U,
Answer: is a set that contains all elements mentioned in a particular
context.
, ◉ The empty set ∅ is not the same as { ∅ }.
Answer: The cardinality of { ∅ } is one since it contains exactly one
element, which is the empty set.
◉ If every element in A is also an element of B, then A is a subset of
B, denoted as A ⊆ B If there is an element of A that is not an element
of B, then A is not a subset of B, denoted as A ⊈ B.
Answer:
◉ Two sets are equal if and only if each is a subset of the other:
Answer: A = B if and only if A ⊆ B and B ⊆ A
◉ If A ⊆ B and there is an element of B that is not an element of A
(i.e., A ≠ B):
Answer: then A is a proper subset of B, denoted as A ⊂ B.
Proper subset is a strictly smaller subset, must be smaller.
◉ Sets of sets.
Answer: A = { { 1, 2 }, ∅, { 1, 2, 3 }, { 1 } }
AND SOLUTIONS GRADED A+
◉ proofs by contrapositive
Answer: proves a conditional theorem of the form p → q by showing
that the contrapositive ¬q → ¬p is true. In other words, ¬c is
assumed to be true and ¬p is proven as a result of ¬q.
Logically equivalent to if p then q
◉ proof by contradiction
Answer: (indirect proof)
starts by assuming that the theorem is false and then shows that
some logical inconsistency arises as a result of this assumption.
Notice not a conditional.
Want to prove Y
Assume not Y
Find a contradiction in X and Not Y
Therefore, claim not not Y.
◉ proof by cases
Answer: A proof by cases of a universal statement such as ∀x P(x)
breaks the domain for the variable x into different classes and gives
,a different proof for each class. Every value in the domain must be
included in at least one class.
◉ Unit 2 sets and functions
Answer:
◉ object in a set are called
Answer: elements
◉ The symbol ∈ is used:
Answer: to indicate that an element is in a set, as in 2 ∈ A
◉ The set with no elements is called the empty set and is denoted by
the symbol ∅.
Answer: The empty set is sometimes referred to as the null setand
can also be denoted by {}. Because the empty set has no elements,
for any element a, a ∉ ∅ is true.
◉ the cardinality of the empty set is:
Answer: zero
◉N
,Answer: The set of natural numbers: All integers greater than or
equal to 0. 0, 1, 2, ...
◉Z
Answer: The set of all integers. ..., -2, -1, 0, 1, 2, ...
◉Q
Answer: The set of rational numbers: All real numbers that can be
expressed as a/b, where a and b are integers and b ≠ 0. 0, 1/2, 5.23, -
5/3
◉R
Answer: The set of real numbers.
◉ The superscript + is used to indicate the positive elements of a
particular set.
Answer: For example, the set R+ is the set of all positive real
numbers, and Z+ is the set of all positive integers.
◉ The universal set, usually denoted by the variable U,
Answer: is a set that contains all elements mentioned in a particular
context.
, ◉ The empty set ∅ is not the same as { ∅ }.
Answer: The cardinality of { ∅ } is one since it contains exactly one
element, which is the empty set.
◉ If every element in A is also an element of B, then A is a subset of
B, denoted as A ⊆ B If there is an element of A that is not an element
of B, then A is not a subset of B, denoted as A ⊈ B.
Answer:
◉ Two sets are equal if and only if each is a subset of the other:
Answer: A = B if and only if A ⊆ B and B ⊆ A
◉ If A ⊆ B and there is an element of B that is not an element of A
(i.e., A ≠ B):
Answer: then A is a proper subset of B, denoted as A ⊂ B.
Proper subset is a strictly smaller subset, must be smaller.
◉ Sets of sets.
Answer: A = { { 1, 2 }, ∅, { 1, 2, 3 }, { 1 } }
