2.1
Write the statements in symbolic form using the symbols ∼,
∨,
and ∧
and the indicated letters to represent component statements.
6. Let s = “stocks are increasing” and i = “interest rates
are steady.”
a. Stocks are increasing but interest rates are steady.
b. Neither are stocks increasing nor are interest rates
steady.
8. Let h = “John is healthy,” w = “John is wealthy,” and s =
“John is wise.”
a. John is healthy and wealthy but not wise.
b. John is not wealthy but he is healthy and wise.
c. John is neither healthy, wealthy, nor wise.
d. John is neither wealthy nor wise, but he is healthy.
e. John is wealthy, but he is not both healthy and wise.
Determine whether the statement forms in 16-24 are logically
equivalent. In each case, construct a truth table and include a sentence
justifying your answer. Your sentence should show that you understand the
meaning of logical equivalence.
22. p ∧ (q ∨ r) and (p ∧ q) ∨
(p ∧
r)
Use De Morgan’s laws to write negations for the statements
in 25-31.
26. Sam is an orange belt and Kate is a red belt.
28. The units digit of 467 is 4 or it is 6.
determine whether the statements in (a) and (b) are
logically equivalent.
45. a. Bob is a double math and computer science major and
Ann is a math major, but Ann is not a double math and computer science major.
b. It is not the case that both Bob and Ann are double math
and computer science majors, but it is the case that Ann is a math major and
Bob is a double math and computer science major.
2.2
Construct truth tables for the statement forms in this
question
6. (p ∨ q) ∨ (∼p ∧ q) → q
11. (p → (q → r)) ↔ ((p ∧ q) → r)
14.
write each of the two statements in symbolic form and
determine whether they are logically equivalent. Include a truth table and a
few words of explanation.
16. If you paid full price, you didn’t buy it at Crown
Books. You didn’t buy it at Crown Books or you paid full price.
20. Write negations for each of the following statements.
(Assume that all variables represent fixed quantities or entities, as
appropriate.)
a.If P is a square, then P is a rectangle.
b.If today is New Year’s Eve, then tomorrow is January.
c.If the decimal expansion of r is terminating, then r is
rational.
d.If n is prime, then n is odd or n is 2.
e.If x is nonnegative, then x is positive or x is 0.
f.If Tom is Ann’s father, then Jim is her uncle and Sue is
her aunt.
g.If n is divisible by 6, then n is divisible by 2 and n is
divisible by 3.
Use truth tables to establish the truth of each statement in
this question
24. A conditional statement is not logically equivalent to
its converse.
27. The converse and inverse of a conditional statement are
logically equivalent to each other.
Rewrite the statements in if-then form.
37. Payment will be made on the fifth unless a new hearing
is granted.
Use the
contrapositive to rewrite the statements in this question in if-then form in
two ways.
43. Doing homework regularly is a necessary condition for
Jim to pass the course.
Use the logical equivalences p →q ≡∼p ∨
q and p ↔ q ≡ (∼p
∨
q) ∧
(∼q
∨
p) to rewrite the given statement forms without using the symbol → or ↔, and (b) use the logical
equivalence p ∨ q ≡∼(∼p
∧
∼q)
to rewrite each statement form using only ∧ and ∼.
49. (p →r) ↔ (q →r)
2.3
Use modus ponens or modus tollens to fill in the blanks in
the arguments of 1-5 so as to produce valid inferences.
1.If √2 is rational, then √2=a/b for some integers a and b.
It is not true that √2=a/b for some integers a and b.
__________________________________________________.
2. If 1 – 0.99999 … is less than every positive real number,
then it equals zero.
___________________________________________________________________________
The number 1 – 0.99999 … equals zero.
3. If logic is easy, then I am a monkey’s uncle. I am not a
monkey’s uncle.
__________________________________________________.
4. If this figure is a quadrilateral, then the sum of its
interior angles is 360°.
The sum of the interior angles of this figure is not 360°.
__________________________________________________.
5. If they were unsure of the address, then they would have
telephoned.
__________________________________________________.
They were sure of the address.
Use truth tables to determine whether the argument forms in this
question are valid. Indicate which columns represent the premises and which
represent the conclusion, and include a sentence explaining how the truth table
supports your answer. Your explanation should show that you understand what it
means for a form of argument to be valid or invalid.
6. P → q
q → p
p ∨ q
7.
P
P → q
∼ q ∨ r
R
8. p ∨ q
P → ∼ q
P →
r
r
9. p ∧
q → ∼
r
P
∨
∼
q
∼
p → p
∼
r
Use symbols to write the logical form of each argument in this
question, and then use a truth table to test the argument for validity.
Indicate which columns represent the premises and which represent the
conclusion, and include a few words of explanation showing that you understand
the meaning of validity.
22. If Tom is not on team A, then Hua is on team B.
If Hua is not on team B, then Tom is on team A.
Tom is not on team A or Hua is not on team B.
