Decide whether the following propositions are true or false.If the claim is valid, supply a short proof, and if the claim is false, provide acounterexample.(a) The arbitrary intersection of compact sets is compact.

Q: Exercise. Is it possible for a metric space (X, p) to contain more than one point and be such that…

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Q: Let A, B and C are sets. Prove that <= is transitive

A: For proving transitivity we need to show that if A&lt;=B and B&lt;=C then A&lt;=C

Q: Give an example of a set in R ^ 2 with the usual topology that is not connected but has a connected…

A: Let's examine the  problem for path-connectedness example of a set in R2 with the usual topology…

Q: Consider two sets in ℝ² with the usual topology. Is the closure of a path-connected set also…

A: We will use knowledge of general topology to solve both the parts of this question. For the first,…

Q: Give two sets that are equal. Prove that they are indeed equal by using the "Axiom of…

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Q: If S is compact and D is in S is a closed set, then D is compact. Use the definition of…

A: Let S is compact and D be a closed subset of S. We are to prove that the set D is compact in S. Now…

Q: Suppose that S is a non-empty finite set. Prove that ? has as many even-sized subsets as it does…

A: Let S be a non-empty set. Let A be the set of even sized subsets of S and let B be the set of odd…

Q: (1) Prove, or come up with as simple a counterexample as you can: For sets A, B, in the context of a…

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Q: Provide a counter example that infinity intersection of open set is open

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Q: Prove that any closed interval [s, t] can be written as the intersection of a collectable collection…

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Q: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

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Q: Prove or Disprove: The set A = {e^x : x × R} is uncountable, where R is the set of real numbers.

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Q: Prove that if A is transitive set, then A+ is also a transitive set.

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Q: 2- Prove that H, is convex set.

A: Let us look at an arbitrary closed half-space defined as H+={x∈ℝn:aTx-b≥0}, representing a half…

Q: Use the definition of compactness to prove that a closed subset Y of a compact set X ⊆ R is compact

A: In this question, we use the definition of compactness to prove that a closed subset Y of a compact…

Q: Can a set and it's Complement be both nowhere dense in R? A Ac=R-A

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Q: Every set S has an open cover.

A: If S is a subset of  R,   Then every set S  will always have an open cover.

Q: B. Let B and C be the subsets of some universal set X, then prove that (C – B)' = BU C'

A: let B and C be the subsets of some universet x1 then prove that (c-B)' = BUC'

Q: If sets X and Y are countable, then prove or disprove that X-Y is uncountable.

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Q: Using the fact that "a countable union of countable sets is countable", show that the statement that…

A: Please see the below picture for detailed solution.

Q: Suppose that A and B are disjoint finite sets. Prove that AUB is finite.

A: Here are given sets A and B are finite sets. Then A=m, B=n  where m, n any fixed positive integer.…

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

A: Solution

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: Give an example of a closed set in R ^ 3 with the usual topology that is not compact

A: Given the usual topology on ℝ3. i.e. τusual=U⊆ℝ3: ∀x∈U, ∃∈&gt;0 such that B∈x⊆U

Q: Xn (Y– Z) = (XnY) – (XnZ) -

A: Determine the truth of the following statement , X∩Y-Z=X∩Y-X∩Z

Q: Decide whether the following statements are true or false. If false, give a counterexample. That is,…

A: a). Given, If A⊆B, then nA≤nB. Since A is a subset of B, therefore all the elements of A are also…

Q: Consider whether the following statements are true or false: a. The intersection of any two convex…

A: To state true or false

Q: 3. Using the definition of open sets, prove that an interval (0, a) is open where a > 0 is an…

A: (3) Definition of open sets: To define open set first we define an open ball For a real number x and…

Q: 4. Use the definition of compactness to prove that a closed subset Y of a compact set X CR is…

A: We will answer the first question since we answer only one question at a time. Please resubmit the…

Q: Use the definition of compact set to prove that the set [0,1] is compact

A: We know that a set E is compact if every open cover of E has a finite subcover. We suppose, that O…

Q: Let X be an infinite set, then any topology on X is also infinite. Select one: O True O False

A: Note: " Since you have asked multiple question. As per our guidelines we are supposed to solve only…

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

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Q: Let X = (6,7), and d be the ususal mortric, d(r, y) = |r - yl Gire ezample of a set ACX, to be not…

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Q: How to Apply the Archimedes'principle to prove that a set is empty

A: Consider the given information. the Archimedes' principle is defined as, Property states that for…

Q: Đ A is a closed set.

A: We prove by using the result that intersection of two closed set is closed.

Q: Decide whether the following propositions are true or false.If the claim is valid, supply a short…

A: Consider the statement: Let A be arbitrary, and let K be compact. Then, the intersection A ∩ Kis…

Q: show that empty set, X is an element of S intersection T in topology

A: Here We have to show that empty set must be in S∩T where S and T are topology on set X.

Q: Construct a family of sets that demonstrates R is not compact

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Q: Prove that the open ball B(p, 8) = {x : ||x – p|| < 8} is a convex set. [Hint: Use the Triangle…

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Q: Given the set T := {(x, y) ∈ R^2 : |xy| Less or equ 1}. Is T a compact set? Show your working. If…

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Q: Give example for following 1/ set is not compact

A: Compact set:- A set S⊆R is called compact “if every sequence in S has a subsequence that converges…

Q: If sets X and Y are uncountable, then prove or disprove that X-Y is uncountable.

A: Here we will prove or disprove the given statement.

Q: 28

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Q: Determine whether each of the following statements is true or false, and explain why.A set is a…

A: A set A is subset of another set B if all elements of the set A are elements of the set B.

Q: Is it true that (A×A)∖(B×B)=(A∖B)×(A∖B) for any two sets A, B? If it is true, prove it. Otherwise,…

A: Explanation of the answer is as follows

Q: Ifs is(finite or infinite) Contable Set then prove that M*(S) ==

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

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Q: prove that every subset of negligible. set is negligible.

A: I have used definition of negligible set.

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.