Let S be a non-empty open subset of R that is bounded above. Show that sup S exists but that max S does not exist.

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: Show that if A is such that m (A)> 0, then there exists a bounded subset BCA such that m'(B) > 0.

A: By properties of Measurable set,         If then  .

Q: Let A be a nonempty subset of R. If a = sup A is finite, show that for each ɛ > 0, there is an a E A…

A: We have to prove the given statement

Q: Prove: If S is compact, and ƒ is continuous on S, then ƒ takes a minimum value some- where in S.

A:

Q: 21. Let A and B be two on-empty bounded sets of positive embers and let С%3D {ху: х€ А and y € B}.…

A:

Q: Let S be a nonempty subset in R". Prove or disprove by a counterexample that Span S = (S+)+

A: We have to prove or disprove by counterexample that Span S=S⊥⊥

Q: Prove that the set (0. 1) = { all xE RIO <x< 1} is uncountable/non-denumerable.

A:

Q: Assume that A and B are nonempty, bounded above, and satisfy B CACR. Show that sup B < sup A.

A: Introduction: The supremum and infimum properties are the very important properties of any subset of…

Q: e) Prove the theorem that states that “S" is a countable set, and there exist a surjection of N onto…

A: Two sets which are one-one and onto, have same cardinality. Let S is a countable set,  S=a1,…

Q: 2. Consider the set A = { n+m² n2+m m eN}. : п, тEN (a) Prove that sup(A) = ∞. (b) Prove that A is…

A: Set and infimum value

Q: 2. Let S be a bounded set in R and. let a < 0. and let aS := {as : s E S). Prove that inf(as) = a…

A: Let S be a bounded set in R and let a&lt;0. And let aS := {as : s belongs to S}. We have to prove…

Q: Find an example of a bounded convex set S in R2 such that its profile P is nonempty but conv P ≠ S.

A:

Q: Let (X, E, µ) be a measure space. Justify through an example that the hypothesis µ(X) < o in…

A:

Q: Prove that the set S = {n² : n e N} is denumerable. %3D

A: We have to show that the given set S is a denumerable set.

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

A:

Q: Show that if A is closed in Y and Y is closed in X, then A is closed in X.

A:

Q: Let S be a nonempty bounded subset of R and let m = sup S. Prove that m is in S iff m = maxS

A: According to the given information, it is required to prove that:

Q: If S, T are nonempty bounded subsets of R with S ⊆ T, then inf T ≤ inf S ≤ sup S ≤ sup T.

A: Since S ⊆ T it follows taht inf T ≤ t, ∀ t ∈ T, then  inf T ≤s , ∀ s ∈S:Therefore inf T is  a lower…

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: Prove that the complement of a dense open subset of R is nowhere dense in R

A: Prove that the complement of a dense open subset of R is nowhere dense in R

Q: prove that an is bounded above.

A: In this question, we prove the sequence  an=(1+1n)n is convergent by given porcedure. i.e.…

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 <…

A:

Q: If S and T are nonempty bounded subsets of the real numbers, show that if ScT Then infT < inf S <…

A: Given ,  S is subset of T

Q: Let A be a subset of a space X and let x ∈ X. Then x ∈ A iff there exists a net in A (that is a net…

A: Given: Let us consider the given, A be a subset of a space X and take x ∈ X.

Q: Exercise 6.40. Let E be a nonempty subset of R that is bounded above. Prove that if a = sup E, then…

A: This is the question of topology.

Q: Show that there exists a set N C (0, 1) that is not Lebesgue measurable.

A:

Q: 6. Let A and B be bounded nonempty subsets of R, and let A+ B:= {a +b: a e A, be B). Prove that…

A:

Q: Show that for any space A there does not exist an injection P(A) → A.

A: To prove that for any space A, there doesn't exist any injection PA→A Let us consider the contrary.…

Q: Consider the subset W = {x € Q : xª > 1} of Q. 2.1 Express W using interval notation. 2.2 Is W…

A: W = x∈ℚ: x4&gt;1 This implies that W is a set of rational numbers 'x', such that x4&gt;1. if x4…

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:

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…

A:

Q: Prove that every finite set S of R" has no accumulation point.

A: Now taking set S Let ? = {?1, ?2, . . . , ?n }.

Q: Let E be a non-empty bounded subset of R. Prove that inf E is unique

A: Given: E is a non-empty bounded subset of ℝ. To show: inf E is unique.

Q: Let S be a nonempty bounded subset ofR. If for every e > 0, there exists an x E S such that x > b -…

A:

Q: Let A CX with the discrete topology. Prove that ô(A) = Ø.

A:

Q: Let S be a nonempty set of real numbers that is bounded from above and let x = sup S. Prove that…

A:

Q: 4. Use the least upper bound property of R to prove that every nonempty subset of R that is bounded…

A: Solution

Q: 3. Let E be a nonempty subset of R that is bounded above, and set U = {ß ER is an upper bound of E}.…

A: Given that Let E be a nonempty subset of R that is bounded above, and set U = { B belongs to R is an…

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: (b) Let # ACR be bounded below. Prove that Inf (A) =-Sup(-A). %3D

A: Let A is non-empty subset of real number . Let A be bounded below then we have to show that  Inf(A)…

Q: 11 Prove that if A CR and |A| > 0, then there exists a subset of A that is not Lebesgue measurable.

A: As per the policy, we are solving a first question. Please repost the question and specify which…

Q: Let there be f: A--&gt;RShow that , if f is bounded in A , then f will also be bounded in B , where…

A: This can be proved as shown below.

Q: Define u* on the power set of R by Jo if A is bounded, 4*(A) = 1 if A is unbounded. Is u* an outer…

A:

Q: Give an example of a collection of compact subsets in R, say (Kn) n=1 to infinity,  such that U n=1…

A: Give an example of a collection of compact subsets in R, say (Kn) n=1 to infinity,  such that U n=1…

Q: |A:k||2||Br:||2) < ||A||F||B||F k=1

A: answer is in next step

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

A:

Q: Prove that S UT = SN T for all sets S and T using the rules for complements of sets.

A: rules of complements x∈A⇔x∉A¯

Q: Find an infinite collection {Kn: n N} of compact sets in R such that the union U K₂ is not compact.…

A: We know the definition of compact sets , A subset S of ℝ is said to be compact if every open cover…

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.