Previously, we listed all 29 topologies on the set X={a,b,c}. However, some of the resulting topological spaces are homeomorphic. Which are homeomorphic? Divide the set of 29 topological spaces into homeomorphism classes, and be sure to justify your choices. There are 9 homeomorphism classes in total. (To justify your choices, explain why the spaces within each class are homeomorphic to each other. Your explanations can be somewhat loose).

Suppose that a decision-maker’s preferences over the set A={a, b, c} are represented by the payoff function u for which u(a) = 0, u(b) = 1, and u(c) = 4.

(a) Are they also represented by the function v for which v(a) =−1, v(b) = 0, and v(c) = 2?

(b) How about the function w for which w(a) =w(b) = 0 and w(c) = 8?

(c) Give another example of a function f:A→R that represents the decision-maker’s preferences.

(d) Is there a function that represents the decision-maker’s preferences and assigns negative numbers to all elements of A?

Let u and v be orthogonal vectors in R3 and let w = 3u + 6v. Suppose that ||u|| = 5 and ||v|| = 4. Find the cosine of the angle between w and v.

A topological space X is zero − dimensional if it has a basis B consisting of open sets which are simultaneously closed. (a) Prove that the set C = {0, 1}^N with the product topology is zero-dimensional. (b) Prove that if (X, d) is a metric space for which |X| < |R|, that is the cardinality of X is less than that of R, then X is zero-dimensional.

Suppose when searching for a large prime, our first step is to sieve by eliminating the first n primes. How large should n be to speed up our search by a factor of 10? This requires careful thought and inventiveness before doing a computation.

How to approach this: Eliminating all odd numbers speeds up our program by a factor of 2, since we've eliminated half of the choices that are composite. If we also immediately disqualify multiples of 3, then we speed up our program by a factor of 3, since we have eliminated 2/3 of all composites. Check that that argument is correct. Now see what happens when you eliminate all of 2, 3 and 5. Again for 2,3,5,7. Now try to figure out the mathematical result and use it to solve for n.

Are the following statements true or false?

1. Let P(n) be the statement "If any string of length n over {a, b} has more a's than b's, then it has two a's in a row". We can prove this statement is true for all n with n ≥ 2 by proving P(2), P(3), and "for all n: P(n) → P(n+2)".

2. Let P(x) be a predicate with one free variable x of type natural. If I prove P(0), "for all x: P(x) → P(x+2)", and "for all x: P(x) → P(x+3)", I may conclude "for all x: P(x)".

2. An industrial sewing machine uses ball bearings that are targeted to have a diameter of 0.75 inches. The lower and upper specification limits under which the ball bearings can operate properly are 0.74 inches and 0.76 inches, respectively.

Past experience has indicated that the actual diameter of the ball bearings is approximately normally distributed with a mean of 0.753 inches and a standard deviation of 0.004 inches.

What is the probability that a ball bearing is:

a. between the target and the actual mean?

b. between the lower specification limit and the target?

c. above the upper specification limit?

d. below the lower specification limit?

e. Of all the ball bearings, 93% of the diameters are greater than what value?

• Document your work clearly and completely. Show your steps
• Use Excel to calculate Z-scores
• Use Appendix Table B.3 Areas Under the Normal Curve from your text to find table probabilities.
• Write complete sentences to answer each question. “The probability of …. is …..”

We can write 13 as a sum of distinct powers of 2: 13 = 8+4+1 = 2 3 +2 2 +2 0 .

• Using strong induction, show every integer can be written as the sum of distinct powers of two.

• Show that every integer has a unique representation as the sum of distinct powers of two. It follows that every integer has a unique binary representation. For instance, 13 is uniquely represented by 1101 in binary

For each of the questions below, write a paragraph using an outside source. Be sure to include a reference of your outside source.

Give a short biography of Alan Turing. In your own words, what is the halting problem?

A dietician is planning a snack package of fruit and nuts. Each ounce of fruit will supply zero units of protein,3 units of carbohydrates, and 2 units of fat, and will contain 40 calories. Each ounce of nuts will supply 2 units of protein,1 unit of carbohydrate, and 4 units of fat, and will contain 50 calories. Every package must provide at least 2 units of protein, at least 7 units of carbohydrates, and no more than 14 units of fat. Find the number of ounces of fruit and number of ounces of nuts that will meet the requirement with the least number of calories. What is the least number of calories?

(a)Let x be the ounces of fruit and y be the ounces of nuts. What is the objective function that must be minimized?

(b)The dietician should use __ ounce(s) of fruit and ___ ounce(s) of nuts. These amounts will have a total of ___ calories

Let p be a prime and d a divisor of p-1. show that the d th powers form a subgroup of U(Z/pZ) of order (p-1)/d. Calculate this subgroup for p=11, d=5; p=17,d=4 ;p=19,d=6

Prove: If A is an uncountable set, then it has both uncountable and countably infinite subsets.