Reference no: EM131061621
1. (a) What is the "Continuum Hypothesis"[CH]? [Please state it precisely in your own words.] (b) Now explain why CH implies that if X, Y, and Z are infinite sets of real numbers, then at least two of them match. (c) Assuming the CH and facts about the sets [0, 1] and IRR (the set of irrationsals) proved in class, explain why [0, 1] must match IRR.
2. (a) If < is a binary relation which linearly orders {a, b, c, d, e}, now many ordered pairs must < contain? Explain. (b) If ~ is an equivalence relation on the same set what is the smallest number of ordered pairs it must contain? Explain.
3. Is it possible for a subset X of a linearly ordered pace (S, <) to be both an open interval and a closed interval? Explain. Why or why not?
4. In the "Square Space" ([0, 1]X[0, 1],< x <) is the subset {(x, x): x belongs to [0, 1]} closed? Why or why not?
5. Let H and K denote closed subsets of an arbitrary linearly ordered space (S, <). If X denotes the union of H and K, and Y denotes the intersection of H and K, prove that both of X and Y are necessarily closed sets.
6. Prove or disprove that if f: (S, <) 3 (S, <) is an isomorphism from a linearly ordered space to itself, then f is a continuous function.
7. Prove that if H is a closed interval and K is a closed set disjoint from H. then there exists an open interval, say U, that contains H and is disjoint from K.. [I would prefer a proof for the general case of an arbitrary linearly ordered space (S, <), but I will accept a proof in the special case of the real line with its usual order if easier to explain.]
8. Prove that if p and q and r are distinct points, then there exist disjoint open sets U and V and W such that p is in U, q is in V, and r is in W.
9. Assume that the space ([0, 1], usual order) is connected (which is true). Use this assumption to prove that the "Square Space" is also connected. (Please say as much as you can even if you can't get a complete proof.)
10. Prove (a) that every constant function from any linearly ordered space to itself is continuous and (b) that the "identity" function defined by f(x) = x for every x in the space is also continuous.
Problem regarding the neil life
: Years ago when Neil was a boy, he was on the debating team at the local high school. He had a slick quality, which helped with the arguments and was often applauded.
|
How much is the net income, after taxes
: how much is the net income, after taxes
|
Give the configuration of the valence shell for germanium
: Give the configuration of the valence shell for germanium. The energy needed to heat a 235 mL cup of water is 59.99 kJ. Using your microwave oven, how many moles of photons are needed to heat a 235 mL cup of water?
|
Identify opportunities to enhance data-analytics efforts
: List the human capital and infrastructure needed to implement data sharing solutions. Include hardware, software, and Cloud storage solutions. Identify opportunities to enhance data-analytics efforts and capabilities.
|
What is the continuum hypothesis
: What is the "Continuum Hypothesis"[CH]? [Please state it precisely in your own words.] Now explain why CH implies that if X, Y, and Z are infinite sets of real numbers, then at least two of them match
|
Reliability or relevance of one of the sources
: For a second peer response, comment on the reliability or relevance of one of the sources listed in a peer post. Consider sharing an additional source or suggestion to your peer.
|
Analyze the final outcomes of the team effort
: Analyze the final outcomes of the team effort in respect to what went wrong or what was done right to facilitate the outcome according to the principles iterated in the Emotional Intelligence text.
|
Determine the amount of federal income tax
: determine the amount of Federal income tax due for the year on a cash basis and also on an accrual basis.
|
Impressionist artists and the surrealist artists
: Identify and explain three major differences between the Impressionist artists and the Surrealist artists and their works. Refer to artists and their works to support your claims. OR Discuss Dali's philosophy and purposes in art as they are reflec..
|