5. Are the following true or false, and prove your answers: (a) If A is compact and B is bounded, then An B is compact. (b) If A is compact and B is closed, then An B is compact. (c) If A is compact and B is bounded, then AUB is compact. (d) If A is compact and B is closed, then AUB is compact.

Q: If A = {x+|x-1| : xER), then which of ONE the following statements is TRUE? A. Set A has a supremum…

A: We use the definition of modulus function to solve this problem.

Q: 3. Consider the Cauchy-Schwarz inequality. That is, for u, v E R": Use some reasoning, or do some…

A:

Q: 39. Which of the following is not correct: O The product of two separable spaces is second countable…

A: First one is not correct.

Q: 3. Use the definition of a convex set to show that if S₁ and S₂ are convex sets in R+", then so is…

A:

Q: Let A and B be sets. Prove that A ≤ B in each of the following. (i) A = {x²: x € Z} and B = Z (ii) A…

A:

Q: 2. Decide whether each of the following sets are finite, denumerable or uncountable. Explain each of…

A:

Q: Suppose A and B are bounded and nonempty subsets of real numbers and 1 E R. In this case: sup(A U B)…

A: Let A,B not empty, bounded subsets of R .Prove thatsup(A∪B)=max{sup(A),sup(B)}. Let: ⇒x∈A∪B⇒x∈A or…

Q: Let A and B be subsets of R which are bounded above. Show that sup (AnB) ≤ min {sup (A), sup (B)}…

A:

Q: 1. For each of the following sets, decide whether they are closed, bounded, compact (both) or…

A:

Q: 2. Let A = [1,3], B = {5}, and C = [4,6]. Write the inequalities (or equalities) that specify each…

A:

Q: A nanodomain is an object studied in the field of mathematical stereotopodynamics. A nanodomain may…

A: A nanodomain is an object studied in the field of mathematical stereotopodynamics. A nanodomain may…

Q: Let A, B be non-empty bounded sets, and A Ç B (A is a proper subset of B). Then sup A < sup B. True…

A: Let A=(1,2] and B=[1,2]

Q: - If C is Convex set ond a GR, then XC is a Convex set

A: Let C be a convex set and x∈R Then we have to show that xC is a convex set. The set xC is defined as…

Q: 7. Sketch the set {(x, y) = R² : ||(x, y) || = 1} when: (a) ||(x, y)=√√√x² + y² (b) ||(x, y) || =…

A:

Q: Is the set {(x1, x2) ∈ R2; 2x1+x2 ≥ 1} convex? Justify your objection.

A: Consider the set      S = {(x1, x2) ∈ R2 : 2x1 + x2 ≥ 1} Let, the points (x1, x2), (x3, x4) be in S.…

Q: Let A and B be nonempty sets of real numbers. If sup A exists then: а. sup (A U B) exists if sup(B)…

A:

Q: Answer the following. Do the best you can entering in mathematical statements into the answer field.…

A:

Q: Justify rigorously the given set is open in R³. (a) A = {(x, y, z) |x>0, y 9}. (b) B = {(x, y, z) |…

A:

Q: What is the interior of S? b. What is the boundary of S? c. Is S open, closed, or neither? All of…

A: a. Interior point is empty b. Boundary of set S is all point of set S c.  Set S is closed.

Q: I need help for number 7. Thanks!

A: (a)Let

Q: True or false (PROVE YOUR ANSWER) :-If A is a set of real numbers that is bounded above and sup A∉…

A: The statement is True.   Proof: Given that A⊆ℝ is bounded above and sup A∉A Let b=sup A Then for any…

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: Prove. For any x, y ∈ R and x < y, ⟨(x, y), ≤⟩ is not well-ordered, where (x, y) is an open interval…

A: To Prove: x,y,≤ is not well ordered, where x,y is an open set with the usual order "≤".

Q: Are the following true or false? If true, prove. If false, give a counterexample. (a) The…

A:

Q: 2. Let A, B and C be sets, and let g: A → B and f: B → C be functions. Prove: (i) If f, g are…

A: Answer given below.

Q: Determine (with argument) whether the following subsets of R are sub- spaces: 1. {(a, b, c) E R³ | a…

A: By Bartleby guidelines I can only answer the first three sub-questions. please repost the question…

Q: Let A, B be bounded subsets of R. Let A+B = {r+y:1 € A, y E B} and AB = {ry : 1 € A, y € B}. Which…

A:

Q: If A p⊂ B, prove that if B is bounded, then A is also bounded. Is the converse true?

A: Note we consider A = A p Let A p⊂B, where B is bounded.

Q: A⊂B, prove that if B is bounded, then A is also bounded. Is the converse true?(

A: Let A⊂B, where B is bounded. Then there exist two real numbers k and K such that k ≤ b ≤ K for all b…

Q: (a) Represent the set {x = (x1, x2) = R² | x1x2 ≥ 1}, as the intersection of some family of…

A: The given set is x=(x1,x2)∈R+2|x1x2≥1. To Find: a) Represent the set x=(x1,x2)∈R+2|x1x2≥1 as the…

Q: B. Let S and T be sets of real numbers, and define S. T to be {x.y:x e Sandy € T}. Show that if S…

A:

Q: a Are ALL the following three pairs in the relation R: (4.2) (1,2) (5,1) b Is 4 a minimum element?…

A: R=4,21,2,5,1a. (1,2) is not the element of R because from Hass diagram 2 is related to 1    here…