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9. The NANP numbers are ten digits in length, and in the format NXX-NXX-XXXX. N can be any 2-9 digit and X can be any digit 0-9. The first three digits are called the numbering plan area (NPA) code. As we know there are 800 area codes available now which means not all of the three-digit combinations can be used as area codes. Some of them are still not used but are reserved for future use. How many possible different 10-digit phone numbers can be created in USA?

10. a. A machine shop has eight screw machines but only three spaces available in the production area for the machine. In how many different ways can the eight machines be arranged in the three spaces available?

b. A quality control randomly selects two of five parts to test for defects. In a group of five parts, how many combinations of two parts can be selected?

10. a. A machine shop has eight screw machines but only three spaces available in the production area for the machine. In how many different ways can the eight machines be arranged in the three spaces available?

b. A quality control randomly selects two of five parts to test for defects. In a group of five parts, how many combinations of two parts can be selected?

5. Let A = {x, y}, B = {1,2}. Find the Cartesian products of A and B: A x B? (Hint: the result will be a set of pairs (a, b) where a ∈ A and b ∈ B).

6. Which if the following sets are equal?

a) {a, b, c, d}

b) {d, e, a, c)

c) {d, b, a, c}

d) {a, a, d, e, c, e}

7. What is the cardinality of each of the following sets?

a) { }

b) { { } }

c) {a, {a}, {a, {a}} }

8. How many different license plates can be made if each plate contains a sequence of three upper case English letters followed by three digits (and no sequences of letters are prohibited, even if they are obscene)?

6. Which if the following sets are equal?

a) {a, b, c, d}

b) {d, e, a, c)

c) {d, b, a, c}

d) {a, a, d, e, c, e}

7. What is the cardinality of each of the following sets?

a) { }

b) { { } }

c) {a, {a}, {a, {a}} }

8. How many different license plates can be made if each plate contains a sequence of three upper case English letters followed by three digits (and no sequences of letters are prohibited, even if they are obscene)?

1. Suppose that A = {2, 4, 6, 8}, B = {2, 4, 8}, and C = {4, 8}. Write out the relationship between A and B, A and C, B and C.

2. Determine whether each of the following pairs of sets are equal.

a) {1, 3, 5, 7} and {3, 1, 7, 5}

b) {{1}} and {1, {1}}

3. Determine whether each of the following statement is true or false:

a) x ∈ {x}

b) {x} ⊆{x}

c) {x} ∈{x}

d) {x} ∈ {{x}}

4. Let A = {b, c, d, f, g}, B = {a, b, c}.

a) Find (A ⋃ B)

b) Find (A ⋂ B)

c) A – B

d) B – A

2. Determine whether each of the following pairs of sets are equal.

a) {1, 3, 5, 7} and {3, 1, 7, 5}

b) {{1}} and {1, {1}}

3. Determine whether each of the following statement is true or false:

a) x ∈ {x}

b) {x} ⊆{x}

c) {x} ∈{x}

d) {x} ∈ {{x}}

4. Let A = {b, c, d, f, g}, B = {a, b, c}.

a) Find (A ⋃ B)

b) Find (A ⋂ B)

c) A – B

d) B – A

Solve this degeneracy problem

Dealer 1 2 3 4 Supply

A 2 2 2 4 1000

B 4 6 4 3 700

C 3 2 1 0 900

Demand 900 800 500 400

Dealer 1 2 3 4 Supply

A 2 2 2 4 1000

B 4 6 4 3 700

C 3 2 1 0 900

Demand 900 800 500 400

Solve this degeneracy problem

11 〖2 〗^8 8 〖6 〗^6 〖2 〗^4 Supply

〖9 〗^10 9 12 9 6 18

7 6 〖3 〗^8 7 7 10

〖9 〗^2 3 5 〖6 〗^2 11 8

Demand 12 8 8 8 4 4

11 〖2 〗^8 8 〖6 〗^6 〖2 〗^4 Supply

〖9 〗^10 9 12 9 6 18

7 6 〖3 〗^8 7 7 10

〖9 〗^2 3 5 〖6 〗^2 11 8

Demand 12 8 8 8 4 4

Solve by NWCM (north-west corner method) and find optimal solution by UV method

3 1 7 4 Supply

2 6 5 9 250

8 3 3 2 350

Demand 200 300 350 400

3 1 7 4 Supply

2 6 5 9 250

8 3 3 2 350

Demand 200 300 350 400

Solve by least cost method and apply UV method to optimize the solution.

19 30 50 13 Supply

70 30 40 60 7

40 10 60 20 10

Demand 5 8 7 15 18

19 30 50 13 Supply

70 30 40 60 7

40 10 60 20 10

Demand 5 8 7 15 18

Solve the given problem by NWCM (north-west corner method), LCM (the least cost method) and VAM (Vogel's approximation method) and find optimal solution by UV method.

11 13 17 14 Supply

16 18 14 10 250

21 24 13 10 300

Demand 200 225 275 400

11 13 17 14 Supply

16 18 14 10 250

21 24 13 10 300

Demand 200 225 275 400

6. Show that ~ (p → q) and p ∧~q are logically equivalent. (Hint: you can use a truth table to prove it or you apply De Morgan law to show the ~(p → q) is p ∧~q.

7.Let p and q be the propositions.

p: I bought a lottery ticket this week.

q: I won the million-dollar jackpot on Friday.

a) Form a tautology using p. Express the tautology in English sentence.

b) Form a tautology using q. Express the tautology in English sentence.

c) Form a contradiction using p. Express the contradiction in English sentence.

d) Form a contradiction using q. Express the contradiction in English sentence.

8. If you have a tautology r and you negate r, what kind of sentence do you get?

a. A tautology

b. A contradiction

c. A sentence that is neither a contradiction nor a tautology

d. You can’t tell—it could be any of (a), (b), or (c).

7.Let p and q be the propositions.

p: I bought a lottery ticket this week.

q: I won the million-dollar jackpot on Friday.

a) Form a tautology using p. Express the tautology in English sentence.

b) Form a tautology using q. Express the tautology in English sentence.

c) Form a contradiction using p. Express the contradiction in English sentence.

d) Form a contradiction using q. Express the contradiction in English sentence.

8. If you have a tautology r and you negate r, what kind of sentence do you get?

a. A tautology

b. A contradiction

c. A sentence that is neither a contradiction nor a tautology

d. You can’t tell—it could be any of (a), (b), or (c).

5. Let p and q be the propositions.

p: You drive over 65 miles per hour.

q: You get a speeding ticket.

Write these propositions using p and q and logical connectives (including negations).

a) You do not drive over 65 miles per hour.

b) You drive over 65 miles per hour, but you do not get a speeding ticket.

c) You will get a speeding ticket if you drive over 65 miles per hour.

d) If you do not drive over 65 miles per hour, then you will not get a speeding ticket.

e) Driving over 65 miles per hour is sufficient for getting a speeding ticket.

f) You get a speeding ticket, but you do not drive over 65 miles per hour.

g) Whenever you get a speeding ticket, you are driving over 65 miles per hour.

p: You drive over 65 miles per hour.

q: You get a speeding ticket.

Write these propositions using p and q and logical connectives (including negations).

a) You do not drive over 65 miles per hour.

b) You drive over 65 miles per hour, but you do not get a speeding ticket.

c) You will get a speeding ticket if you drive over 65 miles per hour.

d) If you do not drive over 65 miles per hour, then you will not get a speeding ticket.

e) Driving over 65 miles per hour is sufficient for getting a speeding ticket.

f) You get a speeding ticket, but you do not drive over 65 miles per hour.

g) Whenever you get a speeding ticket, you are driving over 65 miles per hour.