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Let A and B be finite sets. The properties of set operations, prove that: notation denotes the complement. Let the universal set be U. Usin (AUB) n (AUBc) = A
2. Which of the following sets are not well defined? Explain. a. The set of wealthy schoolteachers b. The set of great books c. The set of natural numbers greater than 100 d. The set of subsets of {1, 2, 3, 4, 5, 6} e. The set {x|x + x and x E N}
(5) Separate N into two disjoint sets: the evens E, and the odds O. Consider the set of Fibonacci ). Prove (n F and En F are infinite sets,6 numbers {1, 1, 2, 3, 5, 8, 13x13 21x21 8x8 Figure 1.10: An interesting geometric proof could use a patterns of the Fibonacci spiral, although there are simpler proofs. the (5) Separate N into two disjoint sets: the evens E, and the odds O. Consider the set of Fibonacci ). Prove...
3. Let E E Lm* (Lebesgue measurable set). Prove that there exist a set G (a countable intersection of open sets), and a set F (a countable union of closed sets) such that F CE C G and m* (F) the Lebesgue measure of a set Hint: The Lebesgue measure can be calculated in terms of open and closed sets m* (E) m* (G), where m* denotes 3. Let E E Lm* (Lebesgue measurable set). Prove that there exist a...
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n E Z. Find [n]. equivalence relation in [N, the equivalence class of 3. We defined a relation on sets A B. Prove that this relation is an (In this view, countable sets the natural numbers under this equivalence relation). exactly those that are are 2. Consider the relation E on Z defined by...
2, For each of these sets. A={3n : n E N), B = {r E R : x2 < 7), and C = {x E R : x < 12), (i) Is the set bounded above? Prove your answer.] ( .] ii) Is the set bounded below? Prove your answer answer the following questions:
Consider two data sets. Set A: n = 5; x = 4 Set B: n = 50; x = 4 (a) Suppose the number 14 is included as an additional data value in Set A. Compute x for the new data set. Hint: x = nx. To compute x for the new data set, add 14 to x of the original data set and divide by 6. (Round your answer to two decimal places.) (b) Suppose the number 14 is...
Given the following sets: S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Even numbers A = {2, 4, 6, 8, 10}; Odd number B = {3, 5, 7, 9}; Natural numbers N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and Prime numbers C = {2, 3, 5, 7} Find the following: a) A ∪ C b) A ∩ N c) A ’ d) B ∩ N e) B ∪ N f) C...
(a + b)(cd)*(a + b) denotes the following set a. {a(cd)nb|n = 1} b. {a(cd)na|n = 1} ? {b(cd)nb/n = 1} c. {a(cd)na|n = 0} ? {a(cd)nb/n = 0} ? {b(cd)na/n = 0} ? {b(cd)nb/n = 0} d. {acndnb|n = 1}
Problem 5. Letf: Z+Zbyn -n. Let D, E S Z denote the sets of odd and even integers, respectively. (a) Prove that fD CE, where D denotes the image of D under f. (b) Is it true that D = E? Prove or disprove. (c) Describe the set f[El. Problem 6. Letf: R R be the function defined by fx) = x2 + 2x + 1. (a) Prove that f is not injective. Find all pairs of real numbers T1,...