Assignment 2

Quantifiers, Rules of Inference, and Proof Techniques

Due Date: Sunday, January 31st 2015, 11:59pm

Requirements

1. (10 pts) Please include your NID (not PID) and full name in your submitted PDF.

2. (10 pts) Be sure your solutions are presented in a clear, readable, and professional way.

Objectives

1. To give students practice with quantifiers.

2. To give students practice with rules of inference.

3. To give students practice with proof techniques.

Problem 1 (15 pts)

Establish the validity of the following arguments using the rules of inference and, if necessary, the laws

of logic. Please note that using truth tables to solve these problems will result in dire consequences

including, but not limited to, being attacked by weasels.

a) ???? ? ???? ? ¬????

?

(?

??????

?????????

)

??????

??????????

? ???? ? ¬????

b) ???? ? ???? ? ????

(

??

??????

???????

???????

?????)

??????

¬??

???????

? ????

c) (???? ? ????) ? ????

?

¬??????

??

????????????????????

? ????

(Continued on the next page.)

Problem 2 (18 pts)

Consider a universe of raccoons and the following definitions of open statements for this universe:

????(????): ???? is an awesome raccoon.

????(????): ???? can juggle chainsaws.

????(????): ???? is riding a bicycle.

????(????): ???? is afraid of waterslides.

Write the following in symbolic form:

a) No awesome raccoon is afraid of waterslides.

b) There exists an awesome raccoon that can juggle chainsaws only when riding a bicycle.

Translate each of the following into an English sentence:

c) ????? ?????(????) ? ¬????(????)?

d) ????? ??????(????) ? ¬????(????)? ? ????(????)?

The following assertions don’t hold in general. Using the open statements defined above, explain why

each of these logical implications fail to hold:

e) ????? ????(????) ? ????? ????(????) ? ????? ?????(????) ? ????(????)?

f) ????? ?????(????) ? ????(????)? ? ????? ????(????) ? ????? ????(????)

Problem 3 (12 pts)

Negate and simplify:

a) ?????????? ?¬????(????) ? ????(????, ????)?

b) ????? ?????(????) ? ????? ¬????(????)?

Problem 4 (6 pts)

Let ????(????) and ????(????) be open statements on the variable ????, for some non-empty universe of discourse.

a) Prove the following:

????? ?????(????) ? ????(????)? ? ????? ????(????) ? ????? ????(????)

b) Does the universe of discourse need to be non-empty for the above assertion to hold?

(Continued on the next page.)

Problem 5 (10 pts)

a) Prove the following:

????? ????(????)

????? ????(????)

????? ?????(????) ? ????(????) ? ????(????)?

????? ?????(????) ? ¬????(????)?

????? ?¬????(????) ? ¬????(????)? ????????????????????????????????????????

? ????? ????(????)

b) Explain why the following is not a valid solution to part (a).

1. ????? ????(????) Premise

2. ????? ????(????) Premise

3. ????(????) Existential Instantiation on (1)

4. ????(????) Existential Instantiation on (2)

5. ????(????) ? ????(????) Conjunction on (3) and (4)

6. ????? ?????(????) ? ????(????) ? ????(????)? Premise

7. ????(????) ? ????(????) ? ????(????) Universal Instantiation on (6)

8. ????(????) Modus Ponens on (5) and (7)

9. ????? ????(????) Existential Generalization on (8)

Problem 6 (10 pts)

Suppose for some integers ????, ????, and ???? that ???? + ???? is odd, ???? + ???? is odd, and ???? + ???? is even.

a) Prove if ???? is odd, then ???? must be even and ???? must be odd.

b) Prove if ????2 is even, then ???? must be odd and ???? must be even.

Assignment 2

Quantifiers, Rules of Inference, and Proof Techniques

Due Date: Sunday, January 31st 2015, 11:59pm

Requirements

1. (10 pts) Please include your NID (not PID) and full name in your submitted PDF.

2. (10 pts) Be sure your solutions are presented in a clear, readable, and professional way.

Objectives

1. To give students practice with quantifiers.

2. To give students practice with rules of inference.

3. To give students practice with proof techniques.

Problem 1 (15 pts)

Establish the validity of the following arguments using the rules of inference and, if necessary, the laws

of logic. Please note that using truth tables to solve these problems will result in dire consequences

including, but not limited to, being attacked by weasels.

a) ???? ? ???? ? ¬????

(???? ? ????) ? ????

???????????????????

? ???? ? ¬????

b) ???? ? ???? ? ????

(????

? ???? ? ????) ? ¬????

???????????????????????

? ????

c) (???? ? ????) ? ????

¬???? ? ????

????????????????????

? ????

(Continued on the next page.)

Problem 2 (18 pts)

Consider a universe of raccoons and the following definitions of open statements for this universe:

????(????):

????(????):

????(????):

????(????):

???? is an awesome raccoon.

???? can juggle chainsaws.

???? is riding a bicycle.

???? is afraid of waterslides.

Write the following in symbolic form:

a) No awesome raccoon is afraid of waterslides.

b) There exists an awesome raccoon that can juggle chainsaws only when riding a bicycle.

Translate each of the following into an English sentence:

c) ????? ?????(????) ? ¬????(????)?

d) ????? ??????(????) ? ¬????(????)? ? ????(????)?

The following assertions don’t hold in general. Using the open statements defined above, explain why

each of these logical implications fail to hold:

e) ????? ????(????) ? ????? ????(????) ? ????? ?????(????) ? ????(????)?

f) ????? ?????(????) ? ????(????)? ? ????? ????(????) ? ????? ????(????)

Problem 3 (12 pts)

Negate and simplify:

a) ?????????? ?¬????(????) ? ????(????, ????)?

b) ????? ?????(????) ? ????? ¬????(????)?

Problem 4 (6 pts)

Let ????(????) and ????(????) be open statements on the variable ????, for some non-empty universe of discourse.

a) Prove the following:

????? ?????(????) ? ????(????)? ? ????? ????(????) ? ????? ????(????)

b) Does the universe of discourse need to be non-empty for the above assertion to hold?

(Continued on the next page.)

Problem 5 (10 pts)

a) Prove the following:

????? ????(????)

????? ????(????)

????? ?????(????) ? ????(????) ? ????(????)?

????? ?????(????) ? ¬????(????)?

?????

?¬????(????) ? ¬????(????)?

????????????????????????????????????????

? ????? ????(????)

b) Explain why the following is not a valid solution to part (a).

1.

2.

3.

4.

5.

6.

7.

8.

9.

????? ????(????)

????? ????(????)

????(????)

????(????)

????(????) ? ????(????)

????? ?????(????) ? ????(????) ? ????(????)?

????(????) ? ????(????) ? ????(????)

????(????)

????? ????(????)

Premise

Premise

Existential Instantiation on (1)

Existential Instantiation on (2)

Conjunction on (3) and (4)

Premise

Universal Instantiation on (6)

Modus Ponens on (5) and (7)

Existential Generalization on (8)

Problem 6 (10 pts)

Suppose for some integers ????, ????, and ???? that ???? + ???? is odd, ???? + ???? is odd, and ???? + ???? is even.

a) Prove if ???? is odd, then ???? must be even and ???? must be odd.

b) Prove if ????2 is even, then ???? must be odd and ???? must be even