WGU C959 DISCRETE MATHEMATICS EXAM
SCRIPT SOLVED QUESTIONS AND CORRECT
ANSWERS GRADED A+
◉ Why are rules of inference useful for establishing the validity of an
argument, especially compared to truth tables? Answer: Truth tables
can become tedious for arguments with many variables, so rules of
inference provide arguments that have already been shown to be valid
and can be reused to prove other arguments valid more efficiently.
(Section 1.11)
◉ What is a logical proof of an argument? Answer: A sequence of
steps, each consisting of a proposition and a justification, that together
establish the validity of the argument. (Section 1.11)
◉ In a logical proof, what justification is given for a step whose
proposition is one of the argument's hypotheses? Answer: The
justification "Hypothesis." (Section 1.11)
◉ In a logical proof, what must be true of a step whose proposition is
not a hypothesis? Answer: The proposition must follow from previous
steps in the proof by applying a law of propositional logic or a rule of
inference. (Section 1.11)
,◉ What information does the justification for a non-hypothesis step in
a logical proof need to include? Answer: The name of the rule or law
being applied and the line numbers of the previous steps to which it is
applied. (Section 1.11)
◉ What must the proposition in the final step of a logical proof be
equal to? Answer: The conclusion of the argument being proven.
(Section 1.11)
◉ When translating an English argument into logic symbols, what is
the first step? Answer: Assigning a propositional variable name to
each individual proposition contained in the argument. (Section 1.11)
◉ Since laws of propositional logic express a two-directional
equivalence, why is it still logically correct to use them within a one-
directional logical proof? Answer: Because using a law of
propositional logic in a proof only requires applying one direction of
the equivalence, so it remains valid even though a proof's overall
conclusion does not imply that its hypotheses are true. (Section 1.11)
◉ Can a law of propositional logic be applied to only part of a larger
compound expression within a step of a logical proof, rather than to
the entire expression? Answer: Yes, laws of propositional logic can be
applied within compound expressions, transforming just a sub-part of
the expression while leaving the rest unchanged. (Section 1.11)
◉ What is required to provide a counterexample showing that an
argument is invalid, in the context of rules of inference? Answer: A
,truth assignment to the variables that makes all of the hypotheses true
while making the conclusion false. (Section 1.11)
◉ What is the modus ponens rule of inference? Answer: From the
hypotheses p → q and p, the conclusion q follows. (Section 1.11)
◉ What is the modus tollens rule of inference? Answer: From the
hypotheses p → q and ¬q, the conclusion ¬p follows. (Section 1.11)
◉ What is the addition rule of inference? Answer: From the
hypothesis p, the conclusion p ∨ q follows, for any proposition q.
(Section 1.11)
◉ What is the simplification rule of inference? Answer: From the
hypothesis p ∧ q, the conclusion p follows. (Section 1.11)
◉ What is the conjunction rule of inference? Answer: From the
hypotheses p and q, the conclusion p ∧ q follows. (Section 1.11)
◉ What is the hypothetical syllogism rule of inference? Answer:
From the hypotheses p → q and q → r, the conclusion p → r follows.
(Section 1.11)
◉ What is the disjunctive syllogism rule of inference? Answer: From
the hypotheses p ∨ q and ¬p, the conclusion q follows. (Section 1.11)
, ◉ What is the resolution rule of inference? Answer: From the
hypotheses p ∨ q and ¬p ∨ r, the conclusion q ∨ r follows. (Section
1.11)
◉ For each definition of an element in the domain, indicate whether
the element defined is particular or arbitrary.
The domain is the set of all integers.
3 is an integer. Hypothesis.
Particular or Arbitrary? Answer: Particular
(Any element defined in a hypothesis is a particular element. 3 is a
particular integer.)
(Section 1.12)
◉ For each definition of an element in the domain, indicate whether
the element defined is particular or arbitrary.
The domain is the set of all employees at a company.
c is an arbitrary employee of the company. Element definition.
Particular or Arbitrary? Answer: Arbitrary
(c is defined to be an arbitrary employee of the company.)
(Section 1.12)
◉ For each definition of an element in the domain, indicate whether
the element defined is particular or arbitrary.
SCRIPT SOLVED QUESTIONS AND CORRECT
ANSWERS GRADED A+
◉ Why are rules of inference useful for establishing the validity of an
argument, especially compared to truth tables? Answer: Truth tables
can become tedious for arguments with many variables, so rules of
inference provide arguments that have already been shown to be valid
and can be reused to prove other arguments valid more efficiently.
(Section 1.11)
◉ What is a logical proof of an argument? Answer: A sequence of
steps, each consisting of a proposition and a justification, that together
establish the validity of the argument. (Section 1.11)
◉ In a logical proof, what justification is given for a step whose
proposition is one of the argument's hypotheses? Answer: The
justification "Hypothesis." (Section 1.11)
◉ In a logical proof, what must be true of a step whose proposition is
not a hypothesis? Answer: The proposition must follow from previous
steps in the proof by applying a law of propositional logic or a rule of
inference. (Section 1.11)
,◉ What information does the justification for a non-hypothesis step in
a logical proof need to include? Answer: The name of the rule or law
being applied and the line numbers of the previous steps to which it is
applied. (Section 1.11)
◉ What must the proposition in the final step of a logical proof be
equal to? Answer: The conclusion of the argument being proven.
(Section 1.11)
◉ When translating an English argument into logic symbols, what is
the first step? Answer: Assigning a propositional variable name to
each individual proposition contained in the argument. (Section 1.11)
◉ Since laws of propositional logic express a two-directional
equivalence, why is it still logically correct to use them within a one-
directional logical proof? Answer: Because using a law of
propositional logic in a proof only requires applying one direction of
the equivalence, so it remains valid even though a proof's overall
conclusion does not imply that its hypotheses are true. (Section 1.11)
◉ Can a law of propositional logic be applied to only part of a larger
compound expression within a step of a logical proof, rather than to
the entire expression? Answer: Yes, laws of propositional logic can be
applied within compound expressions, transforming just a sub-part of
the expression while leaving the rest unchanged. (Section 1.11)
◉ What is required to provide a counterexample showing that an
argument is invalid, in the context of rules of inference? Answer: A
,truth assignment to the variables that makes all of the hypotheses true
while making the conclusion false. (Section 1.11)
◉ What is the modus ponens rule of inference? Answer: From the
hypotheses p → q and p, the conclusion q follows. (Section 1.11)
◉ What is the modus tollens rule of inference? Answer: From the
hypotheses p → q and ¬q, the conclusion ¬p follows. (Section 1.11)
◉ What is the addition rule of inference? Answer: From the
hypothesis p, the conclusion p ∨ q follows, for any proposition q.
(Section 1.11)
◉ What is the simplification rule of inference? Answer: From the
hypothesis p ∧ q, the conclusion p follows. (Section 1.11)
◉ What is the conjunction rule of inference? Answer: From the
hypotheses p and q, the conclusion p ∧ q follows. (Section 1.11)
◉ What is the hypothetical syllogism rule of inference? Answer:
From the hypotheses p → q and q → r, the conclusion p → r follows.
(Section 1.11)
◉ What is the disjunctive syllogism rule of inference? Answer: From
the hypotheses p ∨ q and ¬p, the conclusion q follows. (Section 1.11)
, ◉ What is the resolution rule of inference? Answer: From the
hypotheses p ∨ q and ¬p ∨ r, the conclusion q ∨ r follows. (Section
1.11)
◉ For each definition of an element in the domain, indicate whether
the element defined is particular or arbitrary.
The domain is the set of all integers.
3 is an integer. Hypothesis.
Particular or Arbitrary? Answer: Particular
(Any element defined in a hypothesis is a particular element. 3 is a
particular integer.)
(Section 1.12)
◉ For each definition of an element in the domain, indicate whether
the element defined is particular or arbitrary.
The domain is the set of all employees at a company.
c is an arbitrary employee of the company. Element definition.
Particular or Arbitrary? Answer: Arbitrary
(c is defined to be an arbitrary employee of the company.)
(Section 1.12)
◉ For each definition of an element in the domain, indicate whether
the element defined is particular or arbitrary.
