Let V C(R ) be a nonempty open set satisfying a – b EV whenever a, b e V. Prove that V = R. %D

Q: j) A U A = A k) A U $ = A

A:

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

A:

Q: 1.* Let U C R? be an open and bounded set and u e C²(U) n C(U) satisfy -1 Ди(х) : in U. 1 + |x|

A: Since I have solved similar ones, Let us show that u attains its minimum on the boundary

Q: Suppose that E1 and E2 are measurable sets. Prove that m(E1 U E2) + m(E n E2) = m(E1) + m(E2).

A: We have to prove that the given statement.

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:

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: If S is a subset of T , then u* ( T) sµ* (S).

A:

Q: Let XC Rn convex. Prove that x is an extreme point of X if and only if the set Cis convex, where X =…

A: Let X⊂ℝn be convex . To prove: x is an extreme point of X if and only if the set C is convex, where…

Q: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

A:

Q: * Let Z be with the discrete topology.11 If A = {1,3,5, 7,...}, then Aº %3D %3D A O N O R O

A:

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: In a normed linear space X, prove that the closed ball, ū (x, r) is a convex set for any хE X, and…

A:

Q: Prove by mathematical induction: X n (U-, Y) = U=1(X nY), for V n E Z* – {1} and sets X,Y,,Y2,. Yp-…

A:

Q: * Let R be with the discrete topology. If A = {1, 3,5, 7, .}, then Aº A O N O RO

A:

Q: Let A = {AC R| A is countable} and Az = {A c R| A° is countable} which is a measurable space? (R,…

A:

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: If A and B are nonempty, bounded, and satisfy A c B then sup A < sup B. If sup A < inf B for sets A…

A: a) Given that A,B are non empty sets, and A⊂B The objective is to show that supA≤sup B Since it is…

Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

A:

Q: be any nonempty set and let µ* be an outer measure on P(X) and let A, B e P(X). If at ne of A, B is…

A:

Q: If S is a subset of T , then u* (T) su* (S).

A:

Q: Let c E R and suppose that an → a. Show that can → ca.

A:

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

A:

Q: Let (X, T) be a topological space and A C X. Prove that A =A.

A: PROOF :

Q: Prove that P(Sn T) P(S) n P(T) for any sets S and T

A: Let, S and T be a any set. also, P(S) be any power set of S and P(T) be any power set of T.x∈P(S∩T)…

Q: Let {Aq | a E I} and {B3 | BEJ} be two indexed collections with nonempty index sets I and J. Prove…

A:

Q: et S be a nonempty subset of R and let k E R. Define kS {ks : s E S}. Prove if k > 0, en sup(kS) = k…

A:

Q: an open ball in the set X, with its radius & satisfying -d(x, a), Vx € X at the open ball Be (x)…

A:

Q: Show that E is a measurable set if and only if for every E > 0 there is a closed set F and an open…

A:

Q: A. Show that T is a topology in X and Show that B,(x) is open set with respect to t for all r > 0

A: This is a problem of Topology.

Q: Let X, Y be sets and f : X →Y. Prove f"(A\B) = f"(A) \ f¯"(B). for any A, B CY.

A: Solution:Given:          X, Y    are the Sets and                      f: X→ Y To prove:  f-1(A \…

Q: Let x, y, z E Z be such that both odd, and z is even. Prov

A: Introduction: Even and odd numbers are distinguished by the following characteristics: Even Number:…

Q: Let X be a set with more than one element. (X,t) is a topological space such tha Vxe X,{x} €T then…

A: Given a set X with more than one element.

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

A:

Q: Let X be an uncountable set and let T = {U C X :U = Ø or U¢ is finite} %3D Then is topological space…

A: let X be an uncountable set and T=U⊑X : U=∅ or Uc is finite i.e cofinite topology on X. Then the…

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

A:

Q: Exercise 6. Let E be an infinite non-demumerable set. For every A € P(E), we set m(A) = 0 if A is…

A: Let E be an infinite non-denumerable set. m will be a measure on PE if it is non-negative, which…

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 E and F be measurable sets each having finite measure. Prove that m(EAF) = 0 if and only if m(E)…

A: To prove: The statement "Let E and F be measurable sets each having finite measure. Then, mE∆F=0 if…

Q: 2. Let f : X → Y be onto. For each b e Y, let A, {As : b € Y} is a partition of X. f6]. Prove that

A: Given that f:X→Y is an onto function. Then for every b∈Y there exists x∈X such that  fx=y Also,…

Q: If S is a subset of T, then u*(T) su* (S)

A:

Q: Let {C, : r € [0, 0)} be the family of indexed sets given by Cr {(x, y) E R : a? + y = r²} a)…

A:

Q: 2 Let A be a fixed nonempty subset of set X and I= {Bc X | A Z B}. Show that (X,I) %3D is a matroid.

A:

Q: Let X, Y be two sets. Prove that X x YC P(P(X UY )).

A:

Q: Let the set be w = {(a, b )| a, b E R }, investigate whether the associativity holds for the…

A:

Q: A topological space (X, t) is a T1-space if and only il for each x e X, the singleton set {x} is…

A:

Q: For nonempty sets A, B, and C, let f: A -> B and g: B-> C. If g is onto, then gof is onto.

A:

Q: Let N be with the discrete topology If A=(1,3,5..), then Bd(A)=

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¯