Question 1

Write 1101012 in decimal form.

Accepted Answer

$110101 _ { 2 } = 1 \cdot 2 ^ { 5 } + 1 \cdot 2 ^ { 4 } + 1 \cdot 2 ^ { 2 } + 1 \cdot 2 ^ { 0 } = 32 + 16 + 4 + 1 = 53 _ { 10 }$

Question 2

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

Accepted Answer

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. 
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. E

Question 3

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.

Accepted Answer

The argument has the form
 $p \rightarrow q$
$q$
Therefore $p$,
 which is invalid; it exhibits the converse error.

Question 4

$\text { Find } 10111 _ { 2 } + 1011 _ { 2 }$

Accepted Answer

None

Question 5

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

Accepted Answer

Converse: If Jim is Tom's grandfather, t

Question 6

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.

Accepted Answer

The answer of Consider the following circuit.   
(a)...

Question 7

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

Accepted Answer

Converse: If Jose is Jan's cousin, then

Question 8

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) $- 5 < x < 2$ (where  x  is a particular real number)

Accepted Answer

a. The variable  S  is not undeclared or

Question 9

Are the following statement forms logically equivalent: $p \vee q \rightarrow p$ and $p \vee ( \sim p \wedge q ) ?$ Include a truth table and a few words explaining how the truth table supports your answer.

Accepted Answer

The statement forms are not logically eq

Question 10

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.

Accepted Answer

If someone does not get an ans

Question 11

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.

Accepted Answer

The argument has the form
 @#LAT-DLM& 
@#LAT-DLM&

Question 12

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

Accepted Answer

Sample answers:
Two statement forms are

Question 13

Write 75 in binary notation.

Accepted Answer

The answer of Write 75 in binary notation....

Question 14

$\text { Write the 8-bit two's complement for } 49 \text {. }$

Accepted Answer

@#LAT-DLM&
So the two's complement is @#LAT-DLM&
Check: @#LAT-DLM& and

Question 15

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

Accepted Answer

Sample answers:
For a form of argument t

Question 16

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.

Accepted Answer

Solution 1: The given statements are not

Question 17

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

Accepted Answer

Converse: If Liu is Sue's cous

Question 18

Consider the argument form: 
$p \wedge \sim q \rightarrow r$
$p \vee q$
$q \rightarrow p$
Therefore $r$. 
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.

Accepted Answer

The given form of argument is

Question 19

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?

Accepted Answer

A and B are knights, and C is a knave.
R

Question 20

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.

Accepted Answer

Let p be "12 divides 709,438," q be "3 d

Write 110101₂ in decimal form.

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

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.

$\text { Find } 10111 _ { 2 } + 1011 _ { 2 }$

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

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.

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

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) $- 5 < x < 2$ (where x is a particular real number)

Are the following statement forms logically equivalent: $p \vee q \rightarrow p$ and $p \vee ( \sim p \wedge q ) ?$ Include a truth table and a few words explaining how the truth table supports your answer.

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.

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.

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

Write 75 in binary notation.

$\text { Write the 8-bit two's complement for } 49 \text {. }$

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

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

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?

Speaking Mathematically

The Logic of Quantified Statements

Elementary Number Theory and Methods of Proof

Sequences, Mathematical Induction, and Recursion

Set Theory

Functions

Relations

Counting and Probability

Graphs and Trees

Analyzing Algorithm Efficiency

Regular Expressions and Finite State Automata

Filters

Exam 2: The Logic of Compound Statements

Write 1101012 in decimal form.

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

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.

Find 101112+10112\text { Find } 10111 _ { 2 } + 1011 _ { 2 } Find 101112​+10112​

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

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.

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

Are the following statement forms logically equivalent: p∨q→pp \vee q \rightarrow pp∨q→p and p∨(∼p∧q)?p \vee ( \sim p \wedge q ) ?p∨(∼p∧q)? Include a truth table and a few words explaining how the truth table supports your answer.

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.

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.

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

Write 75 in binary notation.

Write the 8-bit two’s complement for 49. \text { Write the 8-bit two's complement for } 49 \text {. } Write the 8-bit two’s complement for 49.

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

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

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?

Speaking Mathematically

The Logic of Quantified Statements

Elementary Number Theory and Methods of Proof

Sequences, Mathematical Induction, and Recursion

Set Theory

Functions

Relations

Counting and Probability

Graphs and Trees

Analyzing Algorithm Efficiency

Regular Expressions and Finite State Automata

Filters

Write 110101₂ in decimal form.

$\text { Find } 10111 _ { 2 } + 1011 _ { 2 }$

Are the following statement forms logically equivalent: $p \vee q \rightarrow p$ and $p \vee ( \sim p \wedge q ) ?$ Include a truth table and a few words explaining how the truth table supports your answer.

$\text { Write the 8-bit two's complement for } 49 \text {. }$