Math Answers

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

Let S be a nonempty subset of plane 2 \ , it is known that every point ( , ) x y in S
satisfies “if x > 0 , then y > 0 ”. Consider the following properties possibly satisfied
by points( , ) x y in S :
(I) If x ≤ 0 , then y ≤ 0 .
(II) If y ≤ 0 , then x ≤ 0 .
(III) If y > 0 , then x > 0 .
Which of the above properties will have to be satisfied by all points( , ) x y in S?
(a) (II) only
(b) (III) only
(c) (I) and (II)
(d) (I) and (III)
(e) (II) and (III)

How many points lie on plane curves 2 2
+ 1
9 4
x y = and 2 2 ( 1) 1
16 9
x y + − = ?
(a) 1 (b) 2 (c) 3 (d) 4 (e) 0

Given two points A (1,2,3), B (7,6,5), let S be the set of all points in (x, y)–plane
such that X
PA is orthogonal to X
PB, then
(a) S is empty
(b) S contains exactly one point
(c) S contains exactly two points
(d) S is a line segment
(e) S is a circle

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

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

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

Let X represent the weight in pounds of king salmon caught at the mouth of certain river and assume that X possesses a normal distribution with mean 30 and standard deviation 6. Calculate the probability that if a fisherman catches a salmon its weight will be:

a) At least 41 pounds
b) Between 20 and 40 pounds

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).

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.