Show that the set K={a, b} with actions + and - defined tables   is a body whose zero element is a and whose singular element is b.

Q: Exercise 16. Prove the above theorem. Hint: show A ∪ B = A ∪ B − (A ∩ B). Theorem 35…

A: To prove: A∪B=A+B-A∩B

Q: Show that a collection {(-∞,q) C R | q is rational } is a basis for a topology on R.

A: Since for your B part there is no information regarding the  topology homework 2 problem 4. Hence if…

Q: the set {a, b, c, d} that is (a) [BB] reflexive and symmetric (b) not symmetric and not…

A:

Q: Prove that there is no set to which every se Show that for any nonempty set A, there xE Bex belongs…

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

Q: Suppose that A and B are bounded, nonempty subsets of R. Prove that AU B is bounded and that sup(AU…

A: Consider the provided question, Suppose A and B are bounded, nonempty subsets of R.

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

A:

Q: Theorem 6.2. Let C be a compact subset of Rstd· Then C has a maximum point, that is, there is a…

A: We start the proof by assuming there are no such maximum element in C.

Q: 1) In a metric space (M, d), show that both Ø and M are open sets in M. 2) In a metric space (M, d),…

A: Open sets :- Let (M, d) be a metric space then a subset A of M is open if and only if each point of…

Q: ] Prove that a nonempty closed subset of R, if it is bounded from below, has a least element.

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Q: Show that every plane through the origin in R³ may be identified with the null space of an element…

A:

Q: Give an example of a compact set A and a closed set B in R2 such that (conv A) N (conv B) = Ø but A…

A:

Q: Let A be a nonempty subset of B. Is it true that the inclusion mapping from A to B is an injection ?…

A: Let A be a nonempty subset of B. Is it true that the inclusion mapping from A to B is an injection?…

Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U ·. U Bn equal A exactly?…

A:

Q: Given a set L sR", Following are true statements regarding conv L(convex hull of L):

A: In this question, we have to choose the correct option for the convex hull. the set L⊂Rn By using…

Q: 2. Is there a set A that satisfies A = {A}? If yes, exhibit one %3D such. If not, Why not exactly?

A: No, A≠{A} for any set A. Let  A=1,2,3 then number of elements in A are 3. Now,  A=1,2,3 here the…

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…

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Q: Let A1,·.., Am be compact subsets of R. Prove that Aı × · … × Am is a compact subset of Rm.

A: Given, A1,A2, ... , Am are compact subsets of R. We prove A1x A2 x ... x Am is a compact subsets of…

Q: 1) In a metric space (M, d), show that both 0 and M are open sets in M. 2) In a metric space (M, d),…

A:

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: Q3. Let N = {1, 2, 3, ..} equipped with the cofinite topology. Prove that M = {7,14, 21} is a closed…

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Q: Let A1, A2, ., Am be finitely many subsets of R'. Assume that z is a cluster point of UA;. Prove…

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

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Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U... U Bn equal A exactly?…

A:

Q: Give an example of a closed subset S of R^2 such that conv S is not closed

A: An example of a closed subset S of R2 such that conv S is not closed is given below.

Q: Extend the linearly independent Set S={x+1, 2X²} to the space base P₂ (R)

A: We are given a linearly independent set S= {x+1, 2x2} We have to extend S to basis of P2(R).    Let…

Q: Show that R³ is spanned by S = {(1,1,1), (1, –1, 0), (0, 1, – 1)}.

A:

Q: The set S=[1,3) is not compact. Which open cover of S contains a finite subcover?

A: This is the question of Topology.

Q: Show that R³ is spanned by S = {(1,1,1), (1, – 1, 0), (0, 1, – 1)}.

A:

Q: 1) In a metric space (M, d), show that both 0 and M are open sets in M. 2) In a metric space (M, d),…

A: Open sets :- Let (M, d) be a metric space then a subset A of M is open if and only if each point of…

Q: 7. If in an inner product space, (x, u) =(x, v) for all x, show that u = v.

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Q: If A and B are nonempty bounded subsets of R with A C B, prove that sup A < sup B.

A:

Q: Let A be a nonempty set and define B={kx : x∈A}, where k≥0 is fixed. Prove that supB=ksupA.

A: Given A be a nonempty set and define B=kx:x∈A where k≥0 is fixed. We have to prove that supB=ksupA…

Q: Prove that the Joukowski map 1 p(2) : (z + maps ONTO T - (-1, 1)

A:

Q: Q3. Let N = {1, 2, 3, ...} equipped with the cofinite topology. Prove that M = {7,14, 21} is a…

A: Given below the detailed solution

Q: Let ACR be a nonempty bounded set. Prove that sup A is unique.

A:

Q: 7. Show that the set Z of integers is NOT a vector space.

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Q: 8. If a sum P1+··+Pg of projections P,: H →H (H a Hilbert space) is a projection, show that

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Q: Let Z be with the co-finite topology If A = {1,3,5, 7,.. }, then A A O O O

A: As Z is with co-finite topology. Every open neighborhood of an infinite set must contain infinitely…

Q: Theorem 6.17. Every compact subset C of R contains a maximum in the set C, i.e., there is an m E C…

A: To prove that, every compact subset C of ℝ contains a maximum in the set C, that is, there is an m∈C…

Q: 3. Let V be a finite dimensional inner product space. If S = {u, v} is an orthonor- mal set, compute…

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: Exercise 60 Let p and q be two points in a topological space X. Show that there exists an open set U…

A:

Q: Show that the subset (a, b) of R is homeomorphic with R.

A: Recall that every nonempty open interval (a,b) is homeomorphic to the nonempty interval (c,d).

Q: Let the non-empty set S = {x - yv3:x,y ER}, show that S is closed'under multiplication then find the…

A: We have to show that the given set S=x-y3:x,y∈R is closed under multiplication and also find the…

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

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Q: Let X be ametric space. If A and B are each dense open subsets of X then AOB is dense open subset of…

A: First we have shown the intersection is open. Then we use the property that if a set intersect every…

Q: Let KCC be an arbitrary non-empty compact set. BEB(P), 1 ≤p≤0, such that o(B) = K. Construct an…

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: 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: Suppose that X = {x₁,x₂,.,x} is a subset Inner Product space. Which option is correct? a. b. C. d.…

A: Given: Suppose X=x1, x2, x3,…, xn is a subset Inner Product Space. We have to choose the correct…