a) Suppose that E is a set. If there exists a function f from E onto N, then E is at most countable.

Q: Let A be a set. Suppose that f: w→ A is onto A. Prove that A is countable.

A:

Q: 1. Given a relation R on a set A, prove that if R is transitive, then so is R-

A: Transitive means aRb and bRc then aRc

Q: 5. If G1,G2 be two given closed sets in a metric space (X, d) prove that: G1 U G2 is closed.

A: We have to prove given result:

Q: 1. Show that if f is a bijection from a set X onto a set Y then f is a bijection from Y onto X.

A: Please post the multiple questions separately. Here I answered question (1) only as per our policy.

Q: 5. Let E be a set that bounded and c > 0. Prove that sup(cE) = -c inf(-E) %3D

A: This is a problem of Real topology, metric topology.

Q: Let E be a subset of a space X. show titat cl(E) = EUd(E). %3D

A:

Q: [3.12] Let S be a nonempty convex set in R", and let f. R" → R be defined as follows:

A: Let S be a nonempty convex set in Rn, and  Let f:Rn→R be defined as follows:…

Q: If F is a closed bounded set, then every infinite subset S of F has a limit point in F.

A:

Q: Let E CR be a closed set. Prove that aE CE.

A:

Q: Prove or disprove. A closed and bounded subset of e' is compact.

A: PROVE OR DISPROVE :-  A closed and bounded subset of l1 is compact.

Q: A map f:x-y is said to be an open map if for every open set U of X, the set f(u) is open in Y. Show…

A: Given, a map f: X→Y is said to be an open map if for every open set U of X, the set f(U) is open in…

Q: For any set S c R, let 5 denote the intersection of all the closed sets containing S. (a) Prove that…

A: Consider the given information.

Q: 1. Let A be an uncountable set and B C A such that there exists a surjective function f : Z+ → B.…

A:

Q: Let A and B denote subsets of R". (a) Prove that ANBCANB. Give an example to show that equality does…

A: (a) We know that for any two sets A, B A∩B⊆A⇒A∩B¯⊆A¯                        (1) ( Since if X⊆Y then,…

Q: -prove that for any bounded Set E, There is abounded set A That is a countable intersection fopen…

A: The solution is:

Q: 3. Prove that if G is an open set dense in the metric space (S, d) then SG is nowhere dense in S.

A:

Q: Suppose R and S are symmetric relations on a set A. Prove that RoS is symmetric iff RoS = So R.

A:

Q: 1. Prove that for all sets X, Y , if XU Y = Xn Y, then X = Y. %3D

A:

Q: Prove whether the following theorem is true or not : If A is any countably infinite set, B is any…

A:

Q: be an open subset in C and f

A:

Q: Suppose (X, S) is a measurable space, E1, . . . , En are disjoint subsets of X, and c1, . . . , cn…

A: Let's suppose that ϕ:X→R defined by, ϕ=c1XE1+c2XE2+⋯+cnXEn So, this function is well defined as…

Q: The set X is an open set in the ambient metric space . O True O False

A: Solution

Q: If M and N are two norms on space X then prove that M and N are equivalent if and only if a subset…

A: Any two norms M and N on a norm space is called equivalent if  M≤N≤M. in the given question we have…

Q: 4. Let X be any finite set and define the relation R on P(X) by R= {(E,F)||E| = |F\, E, F € P(X)}.

A:

Q: Let B be a set and let A ⊆ B be a subset. Suppose that there exists an injective function f : B → A.…

A: Schroder-Bernstein Theorem- Let A and B be arbitrary sets. Using the concept of cardinality. It is…

Q: Suppose S and T are infinite sets and φ: S→T is a function that is not onto. May we conclude that S…

A:

Q: Let A be a finite set, and suppose that f : A → A is one-to-one. Does f necessarily have to be onto?…

A: The given question is solved below.

Q: Suppose A is a set and R is a reflexive and transitive relation on A. Show that the

A:

Q: 5. Show that the set {e" : n e Z} is countable. (Hint: Use the function f(n) = en and show it is 1-1…

A:

Q: . Let E be a measurable set with m(F np with ISp <∞. Further if Lim (E) then lI f llo = %3D

A:

Q: Let A, BCR be two non-empty bounded sets. Assume that for every x E A, there exists y E B such that…

A:

Q: Let X be an infinite set with the finite closed topology T={S subset of X; X-S is finite}. Then

A: a) If not, then there are non-empty disjoint open sets  such that  . Since they are disjoint, so  .…

Q: Let X = R (or any uncountable set). Define µ* : P(X) → R by { if A is countable µ*(A) = 1 if A is…

A:

Q: 2. Let X be any uncountable set, Prove that: 7. = (GC X: G is countable} U {0} is a topology on X.

A: Since you have posted multiple question... according to company rule we are supposed to answer first…

Q: Show that Let ECR" is measurable set. If µ(E) >0, then E have a non-measurable subset

A: Here, we introduce the notation Gr :=G∩[0, r) + 1-r ∪ G∩[r,1) - r for G⊂[0,1) and r∈R. Note that…

Q: Prove that he interval E = (a, +oo) is an open set in R. (Show all details of your work).

A:

Q: Let X be a nonempty set and fix x E X. Define (1, rE A (A) = 10, 4 A

A:

Q: 3. Prove that the image of a measurable set E under the translation 4 is measurable and m(E+a)…

A:

Q: Suppose f(z) is holomorphic in an open connected set N. Can g(z) = f(z) be holomorphic in N? Explain…

A:

Q: show by an example that a nonempty proper subset of S in the metric space (S, d) can be both open…

A:

Q: Suppose A is an uncountable set and B is a denumerable set. Suppose f:A→B is a function. Prove that…

A:

Q: Let A be dense subset of (X, t)

A: Given: Let A be dense subset of x,Τ and let B be non empty open subset of X. To find: Fill in the…

Q: A relationR is called a partially ordered set if R is: .a reflexive .b antisymmetric .C transitive…

A:

Q: 4.Describe the largest set S on which it is correct to say that f is continuous S (x,y) = In(1–x² –…

A: Given problem:- Describe the largest set S on which it is correct to say that f is continuous C.…

Q: let E be a set that baunded. Prove that Inf E = Sup -E) where R, XE E y

A: To prove for a buonded set E inf(E) = -sup(-E) , where -E = {-x, x belongs to E} , we first prove…

Q: 1. Let X be any nonempty set and let p E X, Prove that : E, {G X:p¢ G}U{X} is a topology on X.

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

Q: 3. Prove if x is an isolated point of a set S, then x E bd S.

A: Since you have posted a multiple question according to guildlines I will solve first question(Q3)…

Q: Let S be a set with a finite number of elements, and let f: S - S be a map _ (a) If fis onto, can If…

A: We are given that S is a set having finite number of elements and f : S--&gt;S be a map.Before we go…

Q: Let F and G be a non-empty families of sets with SG. Show that ng s UG.

A: Given: F and G are two non-empty family of sets with F⊆G, To show: ∩F⊆∪G

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.