If S, T are nonempty bounded subsets of R with S ⊆ T, then inf T ≤ inf S ≤ sup S ≤ sup T.

Q: Suppose that X,Y and Z are subsets of {1, 2, 3, . . . , 10} and |X| = |Y| = |Z| = 7. (i) Prove that…

A: Given that X,Y,Z are subsets of {1,2,3,...,10} |X|=|Y|=|Z|=7. X∪Y≤10 i)X∩Y≥4 X∪Y=X+Y-X∩YX∩Y=X+Y-X∪Y…

Q: Show that if A is such that m (A)> 0, then there exists a bounded subset BCA such that m'(B) > 0.

A: By properties of Measurable set,         If then  .

Q: Show that the subset of R2 given by {(x, y) E R² : xy 1, x > 0} is closed.

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Q: Prove that the set (0. 1) = { all xE RIO <x< 1} is uncountable/non-denumerable.

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Q: 2. Let x be an element of a convex set S. Assume that x1 = x + €ip E S and x2 = x – €2P E S, where p…

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Q: [3.12] Let S be a nonempty convex set in R", and let f. R" → R be defined as follows:

A: Let S be a nonempty convex set in Rn, and  Let f:Rn→R be defined as follows:…

Q: 1. For a subset YC (X,T ) , show that the collection TY ={Y NUJU ET} is closed under finite…

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Q: Find an example of a closed convex set S in R2 such that its profile P is nonempty but convP ≠ S.

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Q: 2. Consider the set A = { n+m² n2+m m eN}. : п, тEN (a) Prove that sup(A) = ∞. (b) Prove that A is…

A: Set and infimum value

Q: 2. Let S be a bounded set in R and. let a < 0. and let aS := {as : s E S). Prove that inf(as) = a…

A: Let S be a bounded set in R and let a&lt;0. And let aS := {as : s belongs to S}. We have to prove…

Q: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

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Q: Let S be a nonempty bounded subset of R and let m = sup S. Prove that m is in S iff m = maxS

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

Q: A and B are bounded subsets of R, then prove that UB and AnB are bounded.

A:

Q: Let S be a non-empty bounded subset of R and let a be an upper bound of S. Show hat sup S = a if and…

A: Follow the steps.

Q: 2. Let S is a subset of R. Prove that S is closed if and only if S contains all of its accumula-…

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Q: Show that if S x T is a closed subset of R2, then S and T are closed subsets of R.

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Q: Prove that if KCR is compact and non-empty, then sup(K) exists and sup(K) E F

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Q: Prove that if A is a nonempty finite set and B is denumerable, then A × B is denumerable.

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Q: . Let S be a bounded set in R and let T be a nonempty subset of S. Show that inf S <inf T< sup T <…

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Q: Prove that if S is a bounded set then S is bounded.

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Q: Let M and N be nonempty bounded subsets of R and let M + N := {m + n : m ∈ M and n ∈ N} . Prove that…

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Q: m) Show that if T;,T; and Ty are topologies on a set X.then r, nnt, is a topology on X.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: If S and T are nonempty bounded subsets of the real numbers, show that if ScT Then infT < inf S <…

A: Given ,  S is subset of T

Q: If X has full rank, so that P = X(X'X)-'X', prove that C(P) = C(X).

A: Given information: It is given that X is a full rank matrix.

Q: If S is a subset of T , then u* (T) su* (S).

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Q: If x is a lower bound for S in R then x must be unique..

A: If x is a lower bound for S in R then x must be unique.

Q: Show that there exists a set N C (0, 1) that is not Lebesgue measurable.

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Q: Consider the following subsets of R equipped with the Euclidean topology: A = { neN} Зп — 1 n+1 ° B…

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Q: Suppose that |x| = n. Find a necessary and sufficient condition on r and s such that (x") C (x*).

A: Given,

Q: Prove that P(Sn T) P(S) n P(T) for any sets S and T

A: Let, S and T be a any set. also, P(S) be any power set of S and P(T) be any power set of T.x∈P(S∩T)…

Q: Suppose that S is a nonempty subset of R and k is an upper bound of S. The number k = sup S if and…

A: Given: S is a non-empty subset of R and k is an upper bound of S. To prove : Number k=sup S if and…

Q: For any measurable nonempty subset of R, L∞ (N) = I¹ (N).

A: Given: Measurable nonempty subset Ω of ℝ. To prove: For any measurable nonempty subset Ω of ℝ,…

Q: Let S be a non-empty open subset of R that is bounded above. Show that sup S exists but that max S…

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Q: Let A, BCR be two non-empty bounded sets. Assume that for every x E A, there exists y E B such that…

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Q: Let x, y, z E Z be such that both odd, and z is even. Prov

A: Introduction: Even and odd numbers are distinguished by the following characteristics: Even Number:…

Q: Let E be a non-empty bounded subset of R. Prove that inf E is unique

A: Given: E is a non-empty bounded subset of ℝ. To show: inf E is unique.

Q: Give an example of a closed subset S of R2 such that conv S is not closed.

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Q: Let S be a nonempty bounded subset ofR. If for every e > 0, there exists an x E S such that x > b -…

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Q: 2. Let f : X → Y be onto. For each b e Y, let A, {As : b € Y} is a partition of X. f6]. Prove that

A: Given that f:X→Y is an onto function. Then for every b∈Y there exists x∈X such that  fx=y Also,…

Q: If S is a subset of T, then u*(T) su* (S)

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Q: Let A and B be nonempty subsets of R such that A is bounded above and B is bounded below and let A –…

A: Let A and B are two non-empty sets of ℝ.Given A is bounded above and B are bounded below sets.Claim:…

Q: 12. If E is a bounded above subset of Z, then A. sup E exists and sup E E E B. sup E exists and sup…

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Q: Let A be a nonempty subset of R that is both bounded above and below and let B be a nonempty subset…

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Q: Give an example of a collection of compact subsets in R, say (Kn) n=1 to infinity,  such that U n=1…

A: Give an example of a collection of compact subsets in R, say (Kn) n=1 to infinity,  such that U n=1…

Q: Let A be a nonempty subset of a bounded set B. Why does inf A and sup A exist? Show that (a) inf B ≤…

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Q: Let A be a nonempty subset of a bounded set B. Why does inf A and sup up A exist? Show that (a) inf…

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Q: Prove that S UT = SN T for all sets S and T using the rules for complements of sets.

A: rules of complements x∈A⇔x∉A¯

Q: Find an infinite collection {Kn: n N} of compact sets in R such that the union U K₂ is not compact.…

A: We know the definition of compact sets , A subset S of ℝ is said to be compact if every open cover…

Q: Let S be a bounded set in R and. let a > 0, and let aS := {as : s E S}. Prove that: i. inf(as) = a…

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Q: Prove that if A C R and t > 0, then |A| = |AN(-t,t)|+|AN (R\(-t,t))|-

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