Using the Internet online services, learn the basic knowledge about wireless wide area network technologies, particularly for mobile phone and data services, and answer the following questions:A large number of consecutive IP addresses are available starting from 198.16.0.0. Suppose that four organizations, A, B, C and D, request 4000, 2000, and 4000, and 8000 addresses, respectively, and in that order. For each of these, give the first IP address assigned, the last IP address assigned, and mark in the notation (for example, 135. 46.56.0/22).Assume that you have purchased the IP network 213.45.32.0/20 from APNIC. Subnet this network so that all of the broadcast domains shown in the below diagram are allocated individual subnets. Create a table with the following headings which lists the subnets that you have created:In a separate table, allocate suitable IP addresses for every router and stations A, P and X. Remember, every port on a router has its own IP address. Allocate your own port numbers for each router. Do not put an IP address on Router 2s port connected to the Internet, as it will get an IP address on this port from your ISP. Write the table out with the following headings:(: the number of stations for each LAN is important for your answer now.)Using the IP addresses from your subnetting, build the above diagram as shown in Packet Tracer. You will create and save a Packet Tracer scenario file in this task. Make sure that all devices and physical connections are represented in your scenario. Give the Internet, indicated in the diagram, the IP address of 131.200.0.0/16, and give Router 2s Internet port the IP address 131.200.1.1/16.Ensure that stations A, P and X can send packets to each other and to the Internet.NOTES:Your work for , saved as a Packet Tracers file.

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

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