WGU C959 Discrete Mathematics I - Exam Questions & Verified Answers
Exclusive or. ⊕ One or the other, but not both.
We can go to the park or the movies.
inclusive or is a: disjunction
Order of operations in absence of parentheses. 1. ¬ (not)
2. ∧ (and)
3. ∨ (or)
the rule is that negation is applied first, then conjunction, then disjunction:
truth table with three variables see pic
2^3 rows
proposition p→q
Ex: If it is raining today, the game will be cancelled.
Converse: q→p
If the game is cancelled, it is raining today.
Contrapositive ¬q → ¬p
If the game is not cancelled, then it is not raining today.
Inverse: ¬p → ¬q
If it is not raining today, the game will not be cancelled.
biconditional p↔q
true when P and Q have the same truth value.
see truth table pic.
free variable ex.
P(x)
the variable is free to take any value in the domain
bound variable ∀x P(x)
bound to a quantifier.
In the statement (∀x P(x)) ∧ Q(x), the variable x in P(x) is bound
the variable x in Q(x) is free.
this statement is not a proposition cause of the free variable.
summary of De Morgan's laws for quantified statements. ¬∀x P(x) ≡ ∃x ¬P(x)
¬∃x P(x) ≡ ∀x ¬P(x)
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, WGU C959 Discrete Mathematics I - Exam Questions & Verified Answers
using a truth table to establish the validity of an argument see pic.
In order to use a truth table to establish the validity of an argument, a truth table is
constructed for all the hypotheses and the conclusion.
A valid argument is a guarantee that the conclusion is true whenever all of the
hypotheses are true.
If when the hypotheses are true, the conclusion is not, then it is invalid.
the argument works if every time the hypotheses (anything above the line) are true,
the conclusion is also true.
hypotheses dont always all need to be true, see example. but every time all the
hypotheses are true, the conclusion needs to be true as well.
rules of inference. see pic.
theorem any statement that you can prove
proof A proof consists of a series of steps, each of which follows logically from
assumptions, or from previously proven statements, whose final step should result
in the statement of the theorem being proven.
the proof of a theorem may make use of axioms: which are statements assumed to be true.
proofs by exhaustion trying everything in the given universe.
proofs by counter example show that one fails.
A counterexample is an assignment of values to variables that shows that a
universal statement is false.
A counterexample for a conditional statement must satisfy all the hypotheses and
contradict the conclusion.
direct proofs used for conditional statements
If p then q
Assume p
Therefore q
proofs by contrapositive 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 (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.
Stuvia 2026-2027 | Latest Version
, WGU C959 Discrete Mathematics I - Exam Questions & Verified Answers
proof by cases 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
object in a set are called elements
The symbol ∈ is used: 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 The empty set is sometimes referred to as the null setand can also be denoted by
denoted by the symbol ∅. {}. Because the empty set has no elements, for any element a, a ∉ ∅ is true.
the cardinality of the empty set is: zero
N The set of natural numbers: All integers greater than or equal to 0. 0, 1, 2, ...
Z The set of all integers. ..., -2, -1, 0, 1, 2, ...
Q 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 The set of real numbers.
The superscript + is used to indicate the positive elements For example, the set R+ is the set of all positive real numbers, and Z+ is the set of
of a particular set. all positive integers.
The universal set, usually denoted by the variable U, is a set that contains all elements mentioned in a particular context.
The empty set ∅ is not the same as { ∅ }. 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.
Two sets are equal if and only if each is a subset of the A = B if and only if A ⊆ B and B ⊆ A
other:
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Exclusive or. ⊕ One or the other, but not both.
We can go to the park or the movies.
inclusive or is a: disjunction
Order of operations in absence of parentheses. 1. ¬ (not)
2. ∧ (and)
3. ∨ (or)
the rule is that negation is applied first, then conjunction, then disjunction:
truth table with three variables see pic
2^3 rows
proposition p→q
Ex: If it is raining today, the game will be cancelled.
Converse: q→p
If the game is cancelled, it is raining today.
Contrapositive ¬q → ¬p
If the game is not cancelled, then it is not raining today.
Inverse: ¬p → ¬q
If it is not raining today, the game will not be cancelled.
biconditional p↔q
true when P and Q have the same truth value.
see truth table pic.
free variable ex.
P(x)
the variable is free to take any value in the domain
bound variable ∀x P(x)
bound to a quantifier.
In the statement (∀x P(x)) ∧ Q(x), the variable x in P(x) is bound
the variable x in Q(x) is free.
this statement is not a proposition cause of the free variable.
summary of De Morgan's laws for quantified statements. ¬∀x P(x) ≡ ∃x ¬P(x)
¬∃x P(x) ≡ ∀x ¬P(x)
Stuvia 2026-2027 | Latest Version
, WGU C959 Discrete Mathematics I - Exam Questions & Verified Answers
using a truth table to establish the validity of an argument see pic.
In order to use a truth table to establish the validity of an argument, a truth table is
constructed for all the hypotheses and the conclusion.
A valid argument is a guarantee that the conclusion is true whenever all of the
hypotheses are true.
If when the hypotheses are true, the conclusion is not, then it is invalid.
the argument works if every time the hypotheses (anything above the line) are true,
the conclusion is also true.
hypotheses dont always all need to be true, see example. but every time all the
hypotheses are true, the conclusion needs to be true as well.
rules of inference. see pic.
theorem any statement that you can prove
proof A proof consists of a series of steps, each of which follows logically from
assumptions, or from previously proven statements, whose final step should result
in the statement of the theorem being proven.
the proof of a theorem may make use of axioms: which are statements assumed to be true.
proofs by exhaustion trying everything in the given universe.
proofs by counter example show that one fails.
A counterexample is an assignment of values to variables that shows that a
universal statement is false.
A counterexample for a conditional statement must satisfy all the hypotheses and
contradict the conclusion.
direct proofs used for conditional statements
If p then q
Assume p
Therefore q
proofs by contrapositive 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 (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.
Stuvia 2026-2027 | Latest Version
, WGU C959 Discrete Mathematics I - Exam Questions & Verified Answers
proof by cases 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
object in a set are called elements
The symbol ∈ is used: 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 The empty set is sometimes referred to as the null setand can also be denoted by
denoted by the symbol ∅. {}. Because the empty set has no elements, for any element a, a ∉ ∅ is true.
the cardinality of the empty set is: zero
N The set of natural numbers: All integers greater than or equal to 0. 0, 1, 2, ...
Z The set of all integers. ..., -2, -1, 0, 1, 2, ...
Q 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 The set of real numbers.
The superscript + is used to indicate the positive elements For example, the set R+ is the set of all positive real numbers, and Z+ is the set of
of a particular set. all positive integers.
The universal set, usually denoted by the variable U, is a set that contains all elements mentioned in a particular context.
The empty set ∅ is not the same as { ∅ }. 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.
Two sets are equal if and only if each is a subset of the A = B if and only if A ⊆ B and B ⊆ A
other:
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