Let S be a non-empty set of real numbers which is bounded above, and suppose that x = sup(S). - (a) Prove that for all € > 0 there exists s € S so that s ≤ (x − €, x].
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![Let S be a non-empty set of real numbers which is bounded above, and
suppose that x = sup(S).
(a) Prove that for all € > 0 there exists s ES so that s = (x − €, x].
(b) Suppose now that x S. Prove that for every e > 0 the set {s ES |
se (x - e, x]} is infinite.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa923b6f-81dd-482c-8885-6de6bc295751%2Fe33c945c-874a-42df-9813-177908c8aedc%2F8k0gdo_processed.jpeg&w=3840&q=75)
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- A set S is called 'denumerable' if there exists a bijection f : N → S. (a) Show that the set N>2 is denumerable because the function g: N → N>2, n > n + 1 is a bijection. (b) Prove that set Z>-3 = {-3, –2, –1,0, 1, 2,3, 4, 5, ...} is denumerable by building a bijec- tive function g: N → Z>-3•Please help. I am having trouble understanding what to do for these questions. Please show your work Thank you5. Prove the statement: Let X and Y be two finite sets with |X|=m and |Y|=n. If there is a one-to-one function f: X → Y, then m≤n.
- Prove that for any a, b R and a-> file:///H:/DISCRETE%20STRUCTURES%20FINAL%2OEXAMS.pc * 4 of 5 0 四 | Let A = {0, 1, 3, 4, 5, 6} and define the relation R and S on A b) as follows: For every (x, y) EA, 1 xRy 5 divides (x2 - y') and 21 47 37 43 46 H B4k 53 K 2 K 96 K 71 KB 01 KE B0 KB 2 KB xSy y – 1 = x 4 KB 4 KB i) List all the ordered pairs (x, y) in the relation R and S. 56KB 56 KB ii) Rn S 52KB iii) SOR (The composite relation R followed by S) 27KB c) The Hasse diagram below represents the partial order R on the set X = {1, 2, 3, 4, 5} 4 NO NAME 2 PluginAlra.. GYMBALS6. Given that -뿔 (금) (규 R(x) 4! 1+. for x E (-, 3), where & is between x and 0, find an upper bound for |R|, valid for all x € [-,), that is independent of r and §. 21 21. (a) (b) (c) (d) Prove or disprove that, for any universal set U and predicates P and Q, [3x = U, P(x) ^ Q(x)] → [3r EU, P(x)) ^ (3x = U, Q(x))] Prove or disprove that, for any universal set U and predicates P and Q, [3r EU, P(x)) ^ (3xU, Q(x))] → [r U, P(x) ^ Q(x)] Prove or disprove that, for any universal set U and predicate P [3r € U, P(x)] → [Vr € U, P(x)] Prove or disprove that, for any universal set U and predicate P [VxU, P(x)] → [3r € U, P(x)]6. Prove that |u. v ≤ uv, and state what condition would imply equality.6.3) If f = u + iv is non-constant and analytic in an open set U, which one of the following are analytic as well? (a) g = u - iv, (b) h = v +iu, (c) k = -u -iv, (d) l= iu - v.1. Let S = (0,7) U 27 +13... Define < on S by a13. (9 points) Let D be the set of finite subsets of positive integers. Let S be the set of all positive integers greate than or equal to 2. Define a function T:S → D as follows: For each integer n ≥ 2, T(n) = the set of all even factors of n. a) Find T(10). b) Find T(17) c) Find T(m), where m is any odd positive integer.Please help me with these questions. I am having trouble understanding what to do. Please show all your work on paper SUBJECT: Discrete mathematics Thank youSEE MORE QUESTIONSRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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