(0) Create a cover page for this quiz that is a whole piece of paper with the following written on it:
Your First and Last Name
Your original Section Number for this course (Sect: 020 or Sect: 021)
Quiz #2
Put nothing else on this cover page. Please do not fill the entire page by writing the above information in a large
font. I need the room on the page for the grading rubric – that’s part of the reason I ask for this page. To that
end, please include the entire page as a pic rather than cropping it.
Number the rest of the problems in this Quiz as indicated by the questions given below. Skip a line
between consecutive problems. Also skip a line between problem parts and subparts.
(1) Draw a truth table to determine whether the logical equivalence below holds. Write a sentence or two to explain
how you use the truth table to inform your answer.
(p → q) → ∼ r ≡ (p → q) ∨ ∼ r
(2) The biconditional, denoted by p ↔ q, is defined by the following logical equivalence:
p ↔ q ≡ (p → q) ∧ (q → p)
In other words, the two-sided arrow is shorthand for the conjunction of an implication (conditional) and its converse.
We read the biconditional p ↔ q as follows: “p if and only if q” or “p is necessary and suficient for q.”
Indicate each of the statements listed below that is logically equivalent to this biconditional statement. (There may
be more than one.)
“Crystal will graduate if and only if she passes all courses required for her degree.”
Hint: “Parse” the given biconditional by assigning a variabe to each of its two components. Parse out the statements
below using the same variables. Use what you know about various logically quivalent forms of p → q and q → p to
decide which of the choices below are L.E. to the biconditional given above. Refer especially to the document “If
P, then Q – A Reference Guide” that appears in the Goode Stuff column in our Reading Page on June 3rd.
(a) If Crystal graduates, then she passes all courses required for her degeree, and, if Crystal does not graduate,
then she does not pass all courses required for her degree.
(b) Crystal graduates only if she passes all courses required for her degree, and, passing all courses required
for her degree is sufficient for Crystal to graduate.
(c) Passing all courses required for her degree is necessary for Crystal to graduate, and, if Crystal does not
pass all courses required for her degree, then she does not graduate.
(d) If Crystal does not graduate, then she does not pass all courses required for her degree, and, Crystal
graduates or she does not pass all courses required for her degree.
(e) Passing all courses required for her degree is sufficient for Crystal to graduate, and, Crystal passes all
courses required for her degree if she graduates.
(3) A tautology is a statement or statement form that is always true. A contradiction is a statement or statement
form that is always false. We denote a tautology by “t” and a contradiction by “c.”
(a) Consider the statement form “p ∨ t.”
(i) Suppose p is True. What is the truth value of p ∨ t?
(ii) Suppose p is False. What is the truth value of p ∨ t?
(iii) Based on your answers to (i) and (ii), do you think p ∨ t ≡ p? Explain.
(iv) Based on your answers to (i) and (ii), do you think p ∨ t ≡ t? Explain.
(b) Consider the statement form “p ∧ t.”
(i) Suppose p is True. What is the truth value of p ∧ t?
(ii) Suppose p is False. What is the truth value of p ∧ t?
(iii) Based on your answers to (i) and (ii), do you think p ∧ t ≡ p? Explain.
(iv) Based on your answers to (i) and (ii), do you think p ∧ t ≡ t? Explain.
(c) Consider the statement form “p ∨ c.”
(i) Suppose p is True. What is the truth value of p ∨ c?
(ii) Suppose p is False. What is the truth value of p ∨ c?
(iii) Based on your answers to (i) and (ii), do you think p ∨ c ≡ p? Explain.
(iv) Based on your answers to (i) and (ii), do you think p ∨ c ≡ c? Explain.
(d) Consider the statement form “p ∧ c.”
(i) Suppose p is True. What is the truth value of p ∧ c?
(ii) Suppose p is False. What is the truth value of p ∧ c?
(iii) Based on your answers to (i) and (ii), do you think p ∧ c ≡ p? Explain.
(iv) Based on your answers to (i) and (ii), do you think p ∧ c ≡ c? Explain.
(4) Work Exercise #12, part (b), at the end of Sec. 2.3 (p. 62) in the text. Use the algorithm discussed in class.
You may wish to refer to the document “Algorithm for Determining Argument Validity/Invalidity” in the Goode
Stuff column for June 3 on the Reading Page.
(5) Read Example 2.3.12 and the definition of a sound argument on page 59 of our text. Continue reading until
you reach the Subsection entitled “Contradictions and Valid Arguments” on that same page.
You should know from reading the text that we have special names for particular sets that we commonly use in
mathematics. For example, the set of all integers is called Z and the set of integers that are strictly positive (greater
than 0) is called Z
+. In other words, Z = {. . . − 3, −2, −1, 0, 1, 2, 3, . . .} and Z

• = {1, 2, 3, . . .}.
  Consider the argument below in which x ∈ Z and y ∈ Z
  +
  y > 0 → x < x + y y > 0
  ∴ x < x + y
  (a) Is this a valid argument form? Refer to class notes and Table 2.3.1. If it is valid, what is the name of that
  form?
  (b) Is this a sound argument? Justify your response.
  (6) Draw the circuit associated with the following Boolean Expression. Be sure and label your AND and OR gates,
  use solder dots when splitting wires, and only use straight wires. Also, draw your circuit so the “juice” flows from
  left to right. If necessary, turn your paper to landscape orientation and use an entire page for your circuit, rather
  than drawing wires that go “backwards.”
  (∼ P ∧ Q ∧ R)_
  (∼ P∧ ∼ Q ∧ R)_
  (∼ P∧ ∼ Q∧ ∼ R)
  (7) Work Exercise #21, parts (a) and (b), at the end of Sec. 2.4 (p. 76) in the textbook.
  (8) Refer to Exercise #29 at the end of Sec. 2.4 (p. 77) in the textbook for this problem.
  (a) Find the I/O Table for the circuit that appears in Exercise #29 (a).
  (b) Find the I/O Table for the circuit that appears in Exercise #29 (b).
  (c) Based on your findings in parts (a) and (b), would you say these are equivalent circuits? Justify your
  response.
  End Quiz #2