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Q: Prove that 1 1 16 ≤ (a+b+c+d) = + = + b + ax (²=²+ + + + + + + ² + 1 = 0) d for all positive numbers…

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Q: Prove R is irreflexive iff R ∩ idA = ∅. If idA={ ∈ A x A |a=b}

A: In the given question we have to prove that a relation R defined on a set A is said to be…

Q: Prove that for any x, y ≥ 0, we have √x - √y ≤ √√x - y). (Hint. Consider the cases x≤y and y ≤ x…

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Q: A is a subset of the real numbers, R.

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Q: Let x and y be distinct real numbers. Prove there is a neighborhood P of x and a neighborhood of y…

A: Answer

Q: d) Suppose X = {1,2, 3} and Y = {5,6}. The number of elements of the set P(X× Y) =

A: Elements in X = 3. Elements in Y = 2 Elements in X × Y = 3 × 2 = 6

Q: If sup A < inf B for sets A and B, then there exists a c€R satisfying a <c<b for all a EA and bEB.

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Q: . Find the sup and inf of the following sets if they exist. a. {xER, x>0|√√x <4} b. (2n + 1 n+1 InEN

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Q: 1. Find the sup and inf of the following sets if they exist. a. {xER, x>0√√x <4} b. (2n + 1 + 1 \nEN…

A: The objective of this question is to find the supremum (the least upper bound) and infimum (the…

Q: Let a, b € R. Show that if a ≤ b₁ for every b₁ > b, then a ≤ b. (Hint: draw a picture if you're…

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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: 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: 3. Prove the triangle inequality: For any real numbers x, y, |x + y ≤ |x| + |y].

A: For any real number  we have to prove

Q: ndicate which of the following statements are true and which are false. Justify your answers as est…

A: Here we have to check given statements are True or False.

Q: Prove: If U is a neighborhood of x0 and U ⊂ V, then V is a neighborhood of x0. Also, if U1 ..., Un…

A: The neighborhood of a point: Let (X, T) be a topological space and x0∈X. A neighborhood of x0 at is…

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 S := a + √2 b + √2 : a, b EN Prove that S is countably infinite.

A: To prove that the set S defined as S={a+2b+2|a,b∈N} is countably infinite, need to establish a…

Q: Suppose E is a bounded, nonvoid set of real numbers. Define |E| to be the set { x: x = E}. Is it…

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Q: Let H = Span{1+x,1+x², 3+x +2x², 2 + x + x²}. what is dimH?

A: Solution

Q: help with this

A: The objective of this question is to prove that the supremum of the sum of two sets A and B is equal…

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

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