Is the following statement true or false? Let A C R. Then a E R is the supremum of A if and only if it is an upper bound for A and Về > 0]x € A:x> a- €.

Q: 1. Let u, ER" prove that lũ ơi <ll||||||

A: we want to shoe that  u→·v→≤u→v→

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: b) Let E = {0,±3,+6,...} be the set of integers that are divisible by 3 and let T = {0, +5, +10,...}…

A: Explanation of the answer is as follows

Q: State the contrapositive of "If f: R→ R is strictly decreasing, then it is injective." Prove or give…

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Q: Let x, y ∈ R such that x 0.001. Prove that |x · y| = |x| · |y|

A: Given: x, y∈ℝ, x&lt;-0.001 and y&gt;0.001   To prove: x·y=x·y

Q: Let a be a non-zero element of Z43. Explain what it means for |a] to be of order w, and hence prove…

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Q: Let D be an integral domain with unity 1. Two non zero elements a, b = D are associates if and only…

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Q: Let A = {x ∈ R| x≠0}, and then define f: A->R by:           f(x) = 6x+10/2x         is f…

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Q: If A is bounded below and B is bounded above, show that inf(A − B) = inf A − sup B

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Q: Prove that if C has infimum 2 then D = {x ∈ R | c ∈ C, x = 3c} has infimum 6

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Q: Find the supremum and infimum of the set {x € R|10/T – I > 0}, if they exist. If they exist, provide…

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Q: Show that |Re(z)| + |Im(z)| ≤ √ 2|z| for any z ∈ C

A: We will use the basic knowledge of the complex number system to answer this question.

Q: Prove that the infimum and the supremum of the interval (a,b) are a and b, respectively

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Q: 1 Assume that Pr[A]=0.45 Pr[B]=0.65, and Pr[A∩B]=0.25  A. Pr[B Compliment ∩ A]= B. Pr[A Compliment…

A: Given, Pr[A]=0.45 Pr[B]=0.65 Pr[A∩B]=0.25

Q: Let x = [1, -3, 2]T. Find ||x||₁, ||x|4, and ||×||∞·

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