whenever a E A and Let A and B be non-empty subsets of R with azb bEB. Prove that sup (A) and inf (13) exist and that sup (A) < Could whenever Sup (A) = inf (B) even though a

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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…

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A: Given that a and b are real numbers and a&lt;b. Prove that (a,b) is an open set

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A: Given that A ∩ B = ø This means P(A ∩ B ) = 0

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Q: Let S be the set S={(x+y) l 2x+3y>=(greater than or equal to) 12} What would be Sc  (S complement)…

A: The given set is: S=x,y∈ℝ2 | 2x+3y ≥12 Cleary S⊂ℝ2.

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Q: Exercise 2.16. Suppose that A and B are countably infinite sets. (A) Show that AUB is countably…

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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: 9. Which of the following are true for all sets A, B, and C. If the statement is true, prove it by…

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Q: (3) Use an element proof to prove that for all sets A, B, C, and D, if An C =0 then (A x B)n (C x D)…

A: The proof of the given statement is given in the explanation part.Explanation:Steps and explanations…

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A: Two statements are given as, A : fx = x2+x logzx is Ox3 B : fx = x logxx is Ωx2 logx where A and B…

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A: According to the given information it is required to prove that

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Q: Prove that the following 2 sets are equal: {4k + 5 : k ∈ Z} = {4k + 1 : k ∈ Z}.

A: Two sets are equal if and only if They have the same element  We are using here if x €A. &lt;=&gt;…

Q: Suppose F and G are families of sets. Prove that (UF)n (UG) ≤ U(F N G) iff VA EFVBE G(An BCU (FNG)).

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Q: If X, Y are finite sets with |X|=4>2= |Y. How many surjective functions f: X→ Y are there? 2 16 8 O…

A: Let us solve above problem

Q: (ii) Determine supremum and the infimum of the set A=(0,10)∩R∪[−2,8]A=(0,10)∩R∪[−2,8] if they exist

A: It is assumed that R denotes the set of real numbers.

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Q: Let S be a set of real numbers such that 5 ∈ S and the sup S > 5. Is it required that S has to have…

A: Let S be a set of real numbers such that 5∈S and thesupS&gt;5.Is it required that S has to have a…

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Q: Provide examples of sets A and B where strict inequality sup(A+B) < sup A+ sup B holds.

A: We have to given an example to prove given inequality:

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A: note : Since you have posted multiple questions, we will provide the solution only to the first…