List the members of the set A={x| x is a real number such that x2=1}, separating them by a comma.

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}.…

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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F open?

A: A set is open if and only if it contains all of its limit points. Also recall that the union of open…

Q: Complete the proof: If A≤B and B≤C, then ACCn(transitive property for set in inclusion) Need to…

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Q: Show that the space (P is complete.

A: The given problem is to show that the space lp is complete. we can use properties of lp spaces to…

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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…

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Q: Find the complement of the set given that U = {x | x I and −3 ≤ x ≤ 9}. (Enter your answers as a…

A: First we write U and the set in roaster form.

Q: 3) Let A be a nonempty set of real numbers that is bounded below. Let -A be the set of all numbers…

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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…

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Q: Let A be a compact set in R. Show that the complement A is open and contains an interval of the form…

A: Solution: Before going into the problem we must know the Theoem named Heine-Borel Theorem.…

Q: Consider the following sets in the dictionarv order. Determine which of these sets are linear…

A: We know that Linear continuum is a linearly ordered set S of more than one element that is densely…

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

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

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Q: Let A and B denote subsets of R". (a) Prove that ANBCANB. Give an example to show that equality does…

A: (a) We know that for any two sets A, B A∩B⊆A⇒A∩B¯⊆A¯                        (1) ( Since if X⊆Y then,…

Q: Let R (the set of real numbers) be equipped with the Euclidean topology, then A, a finite subset of…

A: Every finite set in Euclidean topological space is closed

Q: List the members of these sets, x | x is a real number such that x^2=1.

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Q: What can you say about a set ACR, if every subset of A is closed in R? Which of the following…

A: Given: Every subset of A is closed in ℝ. To do: What conclusion can be given about A.   Conclusion:…

Q: (a) Investigate whether d: RxRR delined as d(z, u) = (- u) is a metrie on the set ol all real…

A: We investigate given distance function is metric on R or not

Q: (b) Let F be a non-empty subset of R. Prove that if OF C F, then F is closed.

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Q: Let A be a subset of Real numbers. Prove that A is Open iff  A Compliment (A') is a closed set.

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Q: Let A be a non-empty, bounded set of positive real numbers. Let B be the set consisting of numbers…

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Q: Let A1 {ACR| A is countable} and A2 = {AC R| A° is countable} which is a measurable space? %3D (IR,…

A: The given sets are countable and every countable set has outer measure zero. Every set which has…

Q: Let C and D be finite sets and let f: C D and g: D C be two functions such that Vy eD f(g(y)) = y…

A: If there exists an one-one map from D to C then "|D| < |C|" or "|C| >|D|".

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

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Q: Use the listing method to express the set S = ((1,3] x {2,4}) n z².

A: Given, S=(1,3]×{2,4}∩ℤ2 We have to express it in the Listing method, As we know, ℤ2=(a,b) where…

Q: Let C and D be finite sets and let f :C → D and g : D →C be two functions such that Vy ED f(g(y)) =…

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Q: Find the complement of the set given that  U = {x | x I and −3 ≤ x ≤ 9}. (Enter your answers as a…

A: First write U and the given set in roster form.

Q: List the elements for the given sets {x|x is a perfect square FROM 1 to 10} {x|x is an integer…

A: Set: A well-defined collection of different objects. For example: Set of natural numbers= {1, 2, 3,…

Q: Let A and B be sets. Fill in the blanks to define precisely what it means for A to be a subset of B.…

A: Let A and B be sets. To fill in the blanks to define precisely what it means for A to be a subset of…

Q: Let A and B be sets. Let X = A n B, Y = A \ X and Z = B\ X. Prove that X n Y = Ø, X n Z = Ø, Y n Z =…

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Q: Consider the number line R with the topology t = {Ø, R, (-0, a); a E R}. Is the set A = (-3, 0]…

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Q: Prove that he interval E = (a, +oo) is an open set in R. (Show all details of your work).

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Q: Let A = {x ER|a° e Q} be a set. Is A countable? Justify the answer by showing the explicit formula…

A: A is set of all real numbers whose cubes are rational.  Now for any real number b, the cube can be…

Q: Assume that A is a σ-algbera of subsets of ℝ. Prove: If A contains all intervals of the form (a,∞),…

A: A σ-algebra on a set X is a collection of subsets of X that is closed under complement and countable…

Q: If X = {u,v} what is the power set of X? (in answering this question, use {} as the null set, and do…

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Q: Let X be a finite set and f: X → X. Prove that f is an injection iff f is a surjection. (Hint: If…

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Q: Suppose {Iα}α∈A is a collection of open intervals that are mutually disjoint: If α1 ≠ α2, then Iα1 ∩…

A: Consider the given information:The collection {{Iα}, where α belongs to A} of open intervals that…

Q: Let A be nonempty bounded subset of real numbers, with u=sup{A}>0. Let A= for all a = 0}, then inf(-…

A: Since you have asked multiple questions in a single request so we will be answering only first…

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

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Q: Let R (the set of real numbers) be equipped with the Euclidean topology, then A, a finite subset of…

A: Solution-

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F closed?

A:

Q: Let 0 denote the empty set. Prove each of the following. (a) (A – B)n (A° U B) = Ø for all sets A…

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Q: Suppose A and B are non-empty bounded subsets of positive real numbers. Define A·B = {r •y:x € A, y…

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Q: let E be a set that baunded. Prove that Inf E = Sup -E) where R, XE E y

A: To prove for a buonded set E inf(E) = -sup(-E) , where -E = {-x, x belongs to E} , we first prove…

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}.…

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Q: Let S be some non-empty collection of real numbers. We know that Vx E S, 3y E S such that x2 y.…

A: In this type of question you need to assume the set of x,y and S that satisfy the condition given in…

Q: Let X be an n-element set. Prove that there are 3n ordered pairs (A, B) of subsets of X such that A…

A: Solution: Let us assume that the subset A has j elements.After fixing A, there are many choices for…

Q: Prove that a set A is connected iff A is and interval.

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Q: Show that the relation R in the set R of real numbers, defined asR = {(a, b) : a ≤ b2} is neither…

A: Given,  relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b2}.