1. For each of these relations on the set {1, 2, 3, 4}, decide whether it is reflexive, whether it is symmetric, whether it is antisymmetric, and whether it is transitive.

1. {(2, 4), (4, 2)}
2. {(1, 2), (2, 3), (3, 4)}
3. {(1, 1), (2, 2), (3, 3), (4, 4)}
4. {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)}

For a) and b) please use the graph representation to determine their properties

For c) and d) please use matrix representation to determine their properties.

1) Define the sets X ={ 1, 2, 3} and Y = {4, 5, 6}. Now, define the relation R from X to Y by xRy if and only if x − y is even. Find all of the elements in R.

2) Define the relation R on ℤ by mRn if and only if 2|(m − n). Show that R is an equivalence relation.

Two large containers A and B of the same size are filled with different fluids. The fluids in containers A and B are maintained at 0° C and 100° C, respectively. A small metal bar, whose initial temperature is 100° C, is lowered into container A. After 1 minute the temperature of the bar is 90° C. After 2 minutes (since being lowered into container A) the bar is removed and instantly transferred into the other container. After 1 minute in container B the temperature of the bar rises 10°. How long, measured from the start of the entire process, will it take the bar to reach 99.7° C? (Round your answer to two decimal places. Assume the final temperature being asked for is reached while the bar is container B.)

Find the pointwise limit f(x) of the sequence of functions fn(x) = x^n/(n+x^n) on [0, ∞). Explain why this sequence does not converge to f uniformly on [0,∞). Given a > 1, show that this sequence converges uniformly on the intervals [0, 1] and [a,∞) for any a > 1.

Solve y^4-4y"=g(t) using variation of parameters.

Verify the Duality Theorem 4.1.9 on each of the following LOPs.
a.
Max. z= −211x1 −189x2 −106x3 −175x4
s.t. 4x1 +6x2 −5x3 +3x4 ≤ 22
x1 +4x3 +9x4 ≤ 18
−3x1 +2x2 −2x3 ≤ 27
5x2 −2x4 ≤ 23
2x1 −x2 −x3 +x4 ≤ 16
7x1 +8x2 −7x3 +4x4 ≤ 12
& x1, x2, x3, x4 ≥ 0
b.
Max. z = x1 + 2x2
s.t. 17x1 + 21x2 ≤ 51
x1 − 4x2 ≤ 12
3x1 + 6x2 ≤ 14
& x1 , x2 ≥ 0

Problem 1. Determine the length, the size, and the minimal distance for each q-ary code C below. How many errors can each code correct? detect?

(a) C = { (0, 0, 0, 0, 0, 0), (0, 1, 1, 1, 1, 0), (1, 0, 0, 0, 0, 1), (1, 1, 1, 1, 1, 1) }. Here q = 2.

(b) C = { (0, 0, 0, 0, 0, 0, 0), (2, 1, 0, 2, 1, 0, 1), (2, 2, 2, 2, 2, 2, 2) }. Here q = 3.

(c) C = { (0, 1, 2, 3, 4), (1, 2, 3, 4, 0), (2, 3, 4, 0, 1), (3, 4, 0, 1, 2), (4, 0, 1, 2, 3) }. Here q = 5.

Problem 2. Assume the code from Problem 1(a) was used in transmission, and the following words were received. Decode each of these words using the nearest neighbour decoding algorithm. (The incomplete decoding version: if there is more than one nearest neighbour, declare an error.)

(a) (0, 1, 0, 0, 0, 0),

(b) (1, 1, 0, 0, 1, 1),

(c) (0, 1, 0, 1, 0, 1),

(d) (1, 1, 0, 0, 0, 0).

Show that the surfaces are tangent to each other at the given point by showing that the surfaces have the same tangent plane at this point. x2 + y2 + z2 − 20x − 16y + 2z + 115 = 0, x2 + y2 + 6z = 17, (5, 4, −4)

Both surfaces have the tangent plane of (answer )

at (5, 4, −4), therefore they are tangent to each other at this point.

Tak took out a loan for $5,000 at 4% interest. To repay the loan he must make a payment of $672.46 at the end of each year for 9 years. How much of his second payment is interest?  

The question we are interested in is this: After he has made payments for 3 years, how much will he still owe?  

Let l and m be two lines in ordinary Euclidean geometry intersecting at point O. Let A, B and C be three distinct points on l and A', B' and C' three distinct points on m (none of them equal to O). Suppose that AB' is parallel to BA', and AC' is parallel to CA'. Prove, using Pappus' theorem in the Extended Euclidean Plane, that BC' and CB' are parallel.

A group of young people were asked how many times they have been stopped and questioned by the police:

                                                Non-White            White

Mean # of Stops               4.95 .99

Varriance 4.11 6.08

Sample size 155 310

We need t-test in order to determine if Non-White and White young people are stopped by the police at the same rate. You should be sure to specify, and evaluate, both the Null and Alternative Hypotheses.

Please show work