Let A and B be nonempty subsets of R such that A is bounded above and B is bounded below and let A – B be defined by A - B= {a – b: a € A and bE B}. Prove that sup(A – B) = sup A – inf B.

Q: Suppose that A₁, ..., An are bounded subsets of R. Prove that A₁ U… U An is bounded and that sup(A₁…

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Q: Let A and B be bounded NON empty sets of R, and let A + B := { a+b: at A, bt B3. Prove that supla…

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Q: Consider the relation R on the set Z defined by a Rb if |a| = |b|. Determine if R is reflexive,…

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Q: Assume that A and B are nonempty, bounded above, and satisfy B CACR. Show that sup B < sup A.

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Q: Consider the set T = {..., −34, −17, 0, 17, 34, ... } with the usual definition of less than (a0).…

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Q: Let A be an open subset of R. Is it true that (A)° = A?

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Q: Show that if f: A→B and E, F are subsets of A then f(E∪F) = f(E)∪f(F) and f(E∩F) ⊆ f(E)∩f(F).

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Q: Consider two sets, A = {1, 2, 3, ..., 100} and B = {3, 5, 7, 9, ... , 201}. Prove that |A| = |B| by…

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Q: We say that a set A is denumerable if there exists a bijective function f: N→ A. Suppose that A is a…

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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: Va Abe B: a <b

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Q: Suppose R and S are symmetric relations on a set A. Prove that RoS is symmetric iff RoS = So R.

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Q: Let (X, T) be a topological space and A C X. Prove that A =A.

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Q: sketch the set {z ∈ C : Re(z) = Im(z + 2 + i)}

A: We have to sketch the set {z∈ℂ:Re(z)=Im(z+2+i)} Let z=x+iy where x,y∈ℝ

Q: Let Y be an infinite subset of a compact set XC R. Prove Y' 0.

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Q: Let D= {1, 2, 3, 4). Give an example of a relation W on D such that W is anti- symmetric but not…

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

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Q: Let T and T 'be two topologies of a set X. Is the family T U T´formed by the openings common to both…

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