Before proceeding, here are standard examples of semialgebras.Examples: why& € 5,₁ = ?(a) Let S = R, and suppose S₁ consists of intervals including Ø, the emptyset:S₁ = {(a, b]: -∞0 ≤a ≤ b ≤∞0}.If I₁, I2 € S₁, then 1₁ 12 is an interval and in S₁ and if I € S₁, then I is aunion of disjoint intervals.I01FIGURE 2.1 IntervalsExercise

Answered: Image /qna-images/answer/27a1a97c-4363-4827-a036-c3b0d3f44be5.jpg

Answered: Before proceeding, here are standard…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Q: State whether the following statements are true or false: 1. Let S be a non-empty subset of the set of real numbers R. Ir s is bounded above, then Sups is exist, but need not to be unique in general. 2. If A = (-5,5) and B = (5,10), then Inf(A + B) = 10 and Sup(A + B) = 15. 3. The closed interval [1,2] has no maximal element. 4. The set of natural numbers N of R is unbounded. 5. The set of real numbers R, has Sup(R) = o and Inf(R) = -co. 6 The set S= (x E RỊ x? - 25 s 0} has Max(S) = 5 and Inf(S) = -5 with no minimal %3D element. 7. The set S = {1+nez*} has Max(S) = 2 and Min(S) = 1. 8. Every bounded set of real numbers R has maximal and minimal elements. 9. The properties (M2) and (M2) of the definition of the metric space are state that the distance from any point to another is never negative, and that the distance from a point to itself is zero. 10. There are many metric functions d: M x M -R that can be defined on a non-empty set M.
each part below, indicate (by making suitable arguments) whether the following sets are open (yes/no), closed yes/no), and bounded (yes/no). Hint: You should probably graph the following sets. The straight line connecting the points (a, b) and (c,d) (including the endpoints) in R² where a < b
Determine the set of real numbers: {1 € R: #= 1 ≥ 4, ∞ #2} x x - by considering the two cases: x2 > 0 and x-2<0 separately.
VI.[15] Let the Universal set be the set R, of all real numbers. [ Hint: Use real number line to represent each set then answer parts a, b and c] A = {x eR|-3
Let Q be the set of rational numbers and R be the set of real numbers, A = {x e R: 5
Q13: Prove or disprove the following: (AUB) - C = (AUC) - (BUC) where A; B; and C are sets.
Let Vi = x ℝ − 1 i ≤ x ≤ 1 i = − 1 i , 1 i for each positive integer i. Find each of the following. (Enter your answers using interval notation.) (a) ∪4i = 1Vi = (b) ∩4i = 1Vi = (c) Are V1, V2, V3, mutually disjoint? Explain. a.) Yes, because the intersection of the sets V1, V2, V3, ... is empty. b.) Yes, because no two of the sets V1, V2, V3, ... have any elements in common. c.) Yes, because the union of the sets V1, V2, V3, ... is empty. d.) No, because no two of the sets V1, V2, V3, ... are disjoint. e.) No, because the sets V1, V2, V3, ... are disjoint. (d) ∪ni = 1Vi = (e) ∩ni = 1Vi = (f) ∪∞i = 1Vi = (g) ∩∞i = 1Vi =
b. For this interval notation, draw its graph and write its equality: (-∞0, 6] 1
Let X + 0, X| 2 2. Then (X, Tina) is not To -space. Select one: OTrue O False
Given X = R (set of real numbers). Determine the T(E) of the following: Note: (a,b) is an interval and not an ordered pair. 8. E= {{1}, {1,2}, {1,3}, {1,4}, {2,3,4}} 9. E= {{1}, (1,2), ({1,2}} 10. E= {(1,3), (-∞, 2), (0,00), (-∞0,4)}
A and B are sets of real numbers defined as follows. A= {w | w<4} B={w | w26} Write An B and A UB using interval notation. If the set is empty, write Ø. (0,0) [0,0) (0,0) AnB =| [0,0) OUO AUB = I - 00 8.
Classify each of the following subsets of R as open, closed, neither, or both. Justify your answer. (a) (-1, 1) U (10, 11) (b) (-∞, 1] Please include interval notation (with a line) for (a) and (b). Thanks!

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