2. Decide whether each of the following sets are finite, denumerable or uncountable. Explain each of your answers. (a) {3m : m E Z} (b) {q € Q : q² E {1,2,3,4}} (c) QU (0,1) (d) Qn (0,1)

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: 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: 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: [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: Prove that the set {a E R: a2 not equal to 6} is open.

A:

Q: 18 What is the size of the space stretched by the set {x, x+x², x³, 1 , x²} ?    a) 2nd  B) 34…

A: Solve the following problem

Q: (12) Give an example of a domain X and predicates p() and q() such that Vx € X, (p(x) → q(x)) is not…

A: Step:-1 Since, we know  P→Q is true if  statement P is true and Q is true or P is false and Q is…

Q: 2. Sketch the following sets. Determine whether each set is open, closed, or neither and whether…

A: We will solve three subparts of the given problem as per the answering guidelines. If you want…

Q: O Show that if Si and S2 are convex sets in R™X", then so is t mxn S = {(x, y1 + y2) | x € R", y1,…

A: Convex set: , If a and b are points in a vector space S. Then the vector space is called convex set…

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

A:

Q: Determine whether is a linearly independent set of vec- 3 11 tors in R3.

A: A set of vectors v1,v2,v3,…,vk in a vector space is said to be linearly independent if…

Q: 1. Define the following sets in extension: (i) {x:N|x< 4• 3 * x} (ii) {x,y:N|x < 6^ y< x•x-y} (iii)…

A:

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…

A:

Q: Show that if Sı and S2 are convex sets in R"+n, then so is their partial sum S = {(x, yı + y2) | x €…

A: We will solve the question number 2.16 as mentioned. Given that S1,S2 are two convex sets in ℝn+m.…

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

A: The solution is:

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

A:

Q: For each of the following sets, state whether the set is finite, countably infinite, or uncountable.…

A: Solution i: Given x∈ℤ+:2x-1=3p for some integer p Since in the given set, it can be placed one to…

Q: 5. Let (X,T) be a discrete space, then Va E X, the set {a} is а. ореn b. closed c. clopen d. is…

A: Let (X, T) be a discrete space.

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

A:

Q: A set SCR" is bounded if there exists k €R such that, for all x E S, || = VE, 0. What can you say…

A:

Q: Suppose that |x| = n. Find a necessary and sufficient condition on r and s such that (x") C (x*).

A: Given,

Q: 5. Suppose X is a closed set, and f: X R. Prove that f is continuous if and only if, for every…

A: This question is about application of integration and differentiable of functions

Q: *12. Let D be a nonempty set and suppose that f: D→R and g: D → R. Define the function f +g:D→R by…

A: Solution 12: a: Since f and g are bounded above on set D. ⇒sup f and sup gexist (completeness axiom)…

Q: 4.4. Let S be a nonempty set. Suppose that to each ordered pair (x, y) E S x S a nonnegative real…

A:

Q: 2- Find the set of lower bounds and the set of upper bounds of the following sets in Q, if it…

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

Q: For any measurable nonempty subset of R, L∞ (N) = I¹ (N).

A: Given: Measurable nonempty subset Ω of ℝ. To prove: For any measurable nonempty subset Ω of ℝ,…

Q: Choose and work 2 problems only. 1. Let E be a bounded infinite set and let x = inf E. Prove : if x…

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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: 1- Find the set of lower bounds and the set of upper bounds of the following sets in R, if it…

A: Since you have posted multiple sunparts of a question so according to our guidelines we will solve…

Q: 2.12 let A and B bounded sets For which there is an ε >o such that la-blze for all ac A, be B .…

A: Consider A and B are bounded sets.

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

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: Identify all nonempty set/s from the following A = {x|x € N, 2x + 7 = 3} B = {x|x € Z, 2x + 7 = 4} C…

A:

Q: [2.13] Let S be a nonempty set in R". Show that S is convex if and only if for each integer k > 2,…

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Q: In a bag of 380 chocolate candies, 40 of them are brown. The candy company claims that 13% of its…

A:

Q: has

A: We have to choose the correct option from the given options according to the given information

Q: Show that the interval [0,1] is a closed set in R

A:

Q: 5. Let (X,T) be a discrete space, then Va E X, the set {a} is а. орen b. closed c. clopen d. is…

A: Our objective is to choose the correct choice

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

A:

Q: Show the following set is convex. x≤10 [(1+(x-10)-²)¹ x>10 Determine a-cut sets of the above set for…

A: To show that the given set is convex: μAx=0                               x≤101+x-10-2-1  x&gt;10…

Q: 1. For what values of k is the null space of A = k -1 a line?

A:

Q: In a bag of 350 chocolate candies, 37 of them are brown. The candy company claims that 13% of its…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub- parts for…

Q: Prove that if A and B are disjoint denumerable sets, then A UB is denumerable.

A:

Q: ave a class boundary that is less than zero?  On the probelem I am working, the class goes from 0-5

A: Class boundaries are the values which help in separating the data points in statistics.

Q: Prove that if A is a nonempty set which is bounded below by 3 then B = {x ∈ R | ∃a ∈ A, b = (1/a)}…

A: Given that A is non empty set which is bounded below by 3. Given that B=x∈R|∃a∈A,b=1a The objective…

Q: 1) Consider the set X = {x E R' / 93x1+34x2 + 26x3 + 18.x4 < 131, x1, x2, 3, X4 E {C (a) Show that…

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

A:

Q: 4.2 Let Fj, be a closed set for k = 1,2, ... ,n in (S, d). Show that U Fk is closed.

A:

Q: Show that if E is any measurable set, then -v (E) < v(E) < v*(E) and |v( E)| < \v|(E).

A: