Let A and B be non-empty and bounded subsets of R such that inf A < inf B. Prove the following: ∃a ∈ A ∀b ∈ B : a < b Hint: Show first that there exists a ∈ A such that a < inf B.

Q: Let B={x∈R|x5∈Q} be a set. Is B countable? Justify the answer by showing the explicit formula for…

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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: Suppose A and B are each bounded above. Define the set A+Bas follows A+B = {a +b: a E A, bE B} We…

A: We know that A number ,α is called G.L.B or infimum of a set of numbers A if and only if α is a…

Q: Let A be a non-empty set of real numbers which has a upremum and let B= {x : - x e A}. Show that B…

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

A: Introduction: The supremum and infimum properties are the very important properties of any subset of…

Q: For nonempty sets A, B, and C, let ƒ : A —> B and g: BC. Prove: If g of is one-to-one, then f is…

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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: Va Abe B: a <b

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Q: Prove that the set {a + b√2 | a, b ∈Q} is countable.

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Q: Let S be a bounded subset of R. The real number a is the supremum of S iff it is an upper bound for…

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Q: Which of the following statements is always true ? Select one: O a. If A, B, and C are sets with AC…

A: Recall countability and uncountability

Q: Let A and B be finite sets such that B = {b1, b2}. Prove that |A – B| < |A|.

A: We are given that A and B are two finite sets, where B= {b1, b2}.We have, |B| = 2, and let |A| =…

Q: Let B be a set and let A ⊆ B be a subset. Suppose that there exists an injective function f : B → A.…

A: Schroder-Bernstein Theorem- Let A and B be arbitrary sets. Using the concept of cardinality. It is…

Q: Given an infinite set X, construct a bijection from X to a proper subset of X.

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Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

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Q: Prove the following: For all sets A, B,and C: A x (BnC) C (A × B)N (A × C).

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Q: A and B are subsets of R (set of reals) Let A be a non empty subset of a bounded set B. why does inf…

A: SOLUTION-

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: For nonempty sets A, B, and C, let ƒ : A —> B and g: B-> C. Prove: If g of is one-to-one, then f is…

A: A function is said to be one-to-one if and only if  implies  for (1) Prove by the direct method…

Q: Let A, B be bounded subsets of R. Which of the following statements follow (tick all that apply)? If…

A: Since A⊆B, every element of A is less than or equal to every element of B. Therefore, supB must be…

Q: Let A and B be any two sets. Prove that Ax B Bx A. Hint: Define f : AxB - Bx A by (a, b) (b, a) for…

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Q: What is the smallest number that can be represented as the sum of four positive squares (not…

A: Consider the given statement. Since we need to find a number, which is the smallest number of sum of…

Q: Let ACR be a bounded non-empty set, and let B := {|x : x € A}. sup B-inf B≤ sup A - inf A (Hint:…

A: Kindly refer to Step 2 for the solution.

Q: real numbers such that a < b. <x < b} is uncountable. <x < b} is countably infinite.

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Q: Prove or disprove: All uncountable measurable subsets of ℝ have positive measure.

A: All uncountable measurable subsets of ℝ have positive measure.