WGU - Discrete Math 1-Introduction to Cryptography
essentials COMPREHENSIVE TEST BANK PRACTICE
EXAM 2026 ACCURATE EXAM APPROVED QUESTIONS
AND CORRECT DETAILED ANSWERS WITH
RATIONALES (CORRECT VERIFIED SOLUTIONS)
CURRENTLY UPDATED 2026-2027
summary of De Morgan's laws for quantified statements.
¬∀x P(x) ≡ ∃x ¬P(x)
¬∃x P(x) ≡ ∀x ¬P(x)
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.
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 denoted by the symbol
∅.
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:
zero
N
The set of natural numbers: All integers greater than or equal to 0. 0, 1, 2,
...
essentials COMPREHENSIVE TEST BANK PRACTICE
EXAM 2026 ACCURATE EXAM APPROVED QUESTIONS
AND CORRECT DETAILED ANSWERS WITH
RATIONALES (CORRECT VERIFIED SOLUTIONS)
CURRENTLY UPDATED 2026-2027
summary of De Morgan's laws for quantified statements.
¬∀x P(x) ≡ ∃x ¬P(x)
¬∃x P(x) ≡ ∀x ¬P(x)
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.
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 denoted by the symbol
∅.
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:
zero
N
The set of natural numbers: All integers greater than or equal to 0. 0, 1, 2,
...
