Deck 2: The Logic of Compound Statements

Question 1

Consider the argument form:
Use the truth table below to determine whether this argument form is valid or invalid. Annotate
the table (as appropriate) and include a few words explaining how the truth table supports
your answer.

Answer 1

The given form of argument is invalid. Row 2 of the truth table shows that it is possible for an argument of this form to have true premises
and a false conclusion.

Question 2

Write the following two statements in symbolic form and determine whether they are logically
equivalent. Include a truth table and a few words explaining how the truth table supports
your answer.
If Sam bought it at Crown Books, then Sam didn't pay full price.
Sam bought it at Crown Books or Sam paid full price.

Answer 2

Solution 1: The given statements are not logically equivalent. Let p be "Sam bought it at Crown
Books," and q be "Sam didn't pay full price." Then the two statements have the following form: The truth tables for these statement forms are Solution 2: The given statements are not logically equivalent. Let p be "Sam bought it at Crown
Books," and q be "Sam paid full price." Then the two statements have the following form: The truth tables for these statement forms are

Question 3

Determine whether the following argument is valid or invalid. Include a truth table and a few
words explaining why the truth table shows validity or invalidity.
If 12 divides 709,438 then 3 divides 709,438.
If the sum of the digits of 709,438 is divisible by 9 then 3 divides 709,438.
The sum of the digits of 709,438 is not divisible by 9.
Therefore, 12 does not divide 709,438.

Answer 3

Let p be "12 divides 709,438," q be "3 divides 709,438," and r be "The sum of the digits of 709,438 is
divisible by 9." Then the given argument has the following form: Row 2 of the truth table shows that it is possible for an argument of this form to have true premises
and a false conclusion. Therefore, the given argument is invalid.

Question 4

Consider the argument form:
Use the truth table below to determine whether this form of argument is valid or invalid.
Annotate the table (as appropriate) and include a few words explaining how the truth table
supports your answer.

Question 5

Consider the following circuit.
(a) Find the output of the circuit corresponding to the input P = 1, Q = 0, and R = 1.
(b) Write the Boolean expression corresponding to the circuit.

Question 6

Write a negation for each of the following statements:
(a) The variable S is undeclared and the data are out of order.
(b) The variable S is undeclared or the data are out of order.
(c) If Al was with Bob on the first, then Al is innocent.
(d) (where x is a particular real number)

Question 7

Question 8

Write the converse, inverse, and contrapositive of "If Ed is Sue's father, then Liu is Sue's
cousin."

Question 9

State precisely (but concisely) what it means for a form of argument to be valid.

Question 10

State precisely (but concisely) what it means for two statement forms to be logically equivalent.

Question 11

Write the form of the following argument. Is the argument valid or invalid? Justify your
answer.
If Ann has the flu, then Ann has a fever.
Ann has a fever.
Therefore, Ann has the flu.

Question 12

Which of the following is a negation for "Jim has grown or Joan has shrunk."

A) Jim has grown or Joan has shrunk.
B) Jim has grown or Joan has not shrunk.
C) Jim has not grown or Joan has not shrunk.
D) Jim has grown and Joan has shrunk.
E) Jim has not grown and Joan has not shrunk.
F) Jim has grown and Joan has not shrunk.

Question 13

Write the converse, inverse, and contrapositive of "If Ann is Jan's mother, then Jose is Jan's
cousin."

Question 14

Write the following two statements in symbolic form and determine whether they are logically
equivalent. Include a truth table and a few words explaining how the truth table supports
your answer.
If Sam is out of Schlitz, then Sam is out of beer.
Sam is not out of beer or Sam is not out of Schlitz.

Question 15

Determine whether the following argument is valid or invalid. Include a truth table and a few
words explaining why the truth table shows validity or invalidity.
If Hugo is a physics major or if Hugo is a math major, then he needs to take calculus.
Hugo needs to take calculus or Hugo is a math major.
Therefore, Hugo is a physics major or Hugo is a math major.

Question 16

Rewrite the following statement in if-then form without using the word "necessary": Getting
an answer of 10 for problem 16 is a necessary condition for solving problem 16 correctly.

Question 17

Write the converse, inverse, and contrapositive of "If Al is Tom's cousin, then Jim is Tom's
grandfather."

Question 18

Which of the following is a negation for "Jim is inside and Jan is at the pool."

A) Jim is inside or Jan is not at the pool.
B) Jim is inside or Jan is at the pool.
C) Jim is not inside or Jan is at the pool.
D) Jim is not inside and Jan is not at the pool.
E) Jim is not inside or Jan is not at the pool.

Question 19

Write the form of the following argument. Is the argument valid or invalid? Justify your
answer.
If 54,587 is a prime number, then 17 is not a divisor of 54,587.
17 is a divisor of 54,587.
Therefore, 54,587 is not a prime number.

Question 20

On the island of knights and knaves, you meet three natives, A, B, and C, who address you
as follows:
A: At least one of us is a knave.
B: At most two of us are knaves.
What are A, B, and C?

Question 21

Write 75 in binary notation.

Question 22

Question 23

Find a circuit with the following input/output table.

Question 24

Question 25

Question 26

Question 27

Write 1101012 in decimal form.