Prove: Let S be an ordered set with the greatest lower bound property, B be a non- empty subset of S, and let B be bounded above. Let Lbe the set of all upper bounds of B. Then a = inf L exits in Sand a = sup B.

Q: Let A be a non-empty subset of R which is bounded below and let B be the subset of R defined by B =…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}.…

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Q: Suppose A and B are each bounded above. Define the set A+Bas follows A+B = {a +b: a E A, bE B} We…

A: We know that A number ,α is called G.L.B or infimum of a set of numbers A if and only if α is a…

Q: Let A be a non-empty set of real numbers which has a upremum and let B= {x : - x e A}. Show that B…

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Q: Let X be an infinite set equipped with the finite closed topology. A finite subset of X is a. closed…

A: The "finite closed topology" describes the closed sets. A subset U is open if and only if U=∅ or X∖U…

Q: Let A be a non-empty subset ofR which is bounded below and let B be the subset of R defined by B =…

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Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}.…

A: We have, A=set of positive numbers  A is bounded above set so, for all  x∈A ∃ K∈ℝ such that x≤K We…

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x € R: 1/x e A}. 1…

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Q: Let S be a non-empty bounded subset of R and let a be an upper bound of S. Show hat sup S = a if and…

A: Follow the steps.

Q: 21. Let A and B be two non-empty bounded sets of positive numbers and let C = {xy : x e A and y e…

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Q: Let W be a nonempty subset of R that is bounded above, and define the set B as follows B= {N ER:N is…

A: Let W be a non-empty subset of ℝ is bounded above, and define the set B as follows, B=Ω∈ℝ: Ω is an…

Q: Let X be an ordered set, and let A be a subset of X.Consider A as an ordered set, where the order on…

A: Given that X be an ordered set and let A⊂X, A is an ordered set. Y⊂X and Y is convex  in X. The…

Q: Let u* be an outer measure on P(X) and M be the collection of all measurable sets in P(X). Let (An)…

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Q: 23. Let A be a non-empty set of positive numbers, which is bounded above and let B = {xe R: 1/xe A}.…

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Q: . Let S be a bounded set in R and let T be a nonempty subset of S. Show that inf S <inf T< sup T <…

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Q: 23. Let A be a non-empty set of positive numbers, which is bounded above and let B = {xe R: 1/xe A}.…

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Q: Let S be an ordered set.  Let B ⊂ S be bounded (above and below). Let A ⊂ B be a nonempty subset.…

A: We can prove this by the definition of upper and lower bound

Q: Let M and N be nonempty bounded subsets of R and let M + N := {m + n : m ∈ M and n ∈ N} . Prove that…

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Q: Let A be a non-empty, bounded set of positive real numbers. Let B be the set consisting of numbers…

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Q: Let A and B be nonempty sets of real numbers. If sup A exists then: a. sup (A U B) exists if sup(B)…

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Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}.…

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Q: 2) Let A be a finite set and a EA.Lf a EA is not a maximal element of A, then a has an îmmediate SUC…

A: Given A is a finite set. a∈A is not a maximal element of A. Now we have to prove 'a' has an…

Q: Let A and B be bounded nonempty subsets of R, and let A + B:= {a +b: a E A, be B). Prove that sup(A…

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Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x € R: 1/x e A}. 1…

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Q: The Cantor set is closed .A O dense .B O open .C O neither open nor closed.D O

A: Closed

Q: Let (2, F,µ) be a measure space. (a). Let A, B E F such that A C B. Prove that u(A) sH(B). (b) Let…

A: Given that Ω,F,μ be a measure space. We know that Ω,F,μ is a measure space if: μϕ=0…

Q: Let A, B Be two subsets af R, such that B is bounded and YaE A ]bEB such that show that: a sb O A is…

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Q: x = sup(A) and if a ɛ A, with x#a , then which of the following statements is Let true? O There is…

A: Given that x is sup(A) . We have to check which of the following statement is true or false.

Q: Help with #6.27 please.

A: To prove that the intersection of any number of closed sets in E is closed.

Q: Let A be a non-empty set of positive numbers, which is unded above and let B = {x e R: 1/xe A}. 1…

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Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}. 1…

A: Remark: Completeness property of R: Every bounded above non-empty subset of R has a least upper…

Q: Prove that if Gn is an open set which is dense in the complete metric space (S,d) for n = 1,2,...…

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Q: Let (E), be a countable disjoint collection of measurable sets. Prove that for any set A, 00 m* (AnU…

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Q: Let $X$ be a set and assume that p E X. Let T, be the collection (Ø, X, and all subsets of X that…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Prove that sup(A) =1, inf(A) = . 5. (i) Let A, BCR be sets which are bounded above, such that AC B.…

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Q: Let {An : n E N} be the indexed collection of sets defined by 1 2, 2+ 2n An Prove that: nMeN An = Ø

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Q: let A Set bounded below, ילC erove that C. nf A inf C A %3D

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Q: Let A be a non-empty set and suppose that f : A → Rand g : A → R. Define f + g : A → R by ( ƒ +…

A: we take f(x)=-1   x∈Q∩A1       x∈Qc∩A and g(x)=2   x∈Q∩A-2   x∈Qc∩A(f+g)(x)=f(x)+g(x)=1   x∈Q∩A-1…

Q: Let A be nonempty bounded subset of real numbers, with u=sup{A}>0. Let A= for all a = 0}, then inf(-…

A: Since you have asked multiple questions in a single request so we will be answering only first…

Q: Let x= sup (A) and if a E A, with x# a then which of the following statements is true? There is rEA…

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Q: 3. Let a, b e R such that a < b. Prove that the open interval (a, b) is an open set in R.

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Q: Let (S,≼) be a linearly ordered set with the Least-Upper-Bound Property. Let A and B be non-empty…

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Q: 23. Let A be a non-empty set of positive numbers, which is ounded above and let B= {xe R: 1/x e A}.…

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Q: Determine if the two sets are dense in R, nowhere dense in R or somewhere in between: A=Q⋂[0,5]…

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Q: Prove that a set S⊂R is everywhere dense iff it intersects every non-empty open sets in R. Prove…

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Q: Let A be a nonempty subset of R that is both bounded above and below and let B be a nonempty subset…

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Q: 5. Let S be a nonempty subset of R that is bounded above, with upper bound b. Prove that b = sup(S)…

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Q: Let {Fa}aeA be a collection of closed subsets of R. (a) Prove, from the definition of closed, that…

A: Given that Fαα∈A is the collection of closed subsets of ℝ. The objective is: To prove that ∩α∈AFαis…

Q: Let A be a nonempty subset of R that is bounded below. Define the set B={b: b is a lower bound of…

A: Given A is nonempty subset of ℝ. Also, A is bounded below. Greatest lower bound property: The…

Q: 2) let A be a finite set and a EA.Lf a EA is not a maximal element of A,then a has an îmmediate suc…

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Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.