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 E Z C. sup E exists and sup E EZ but sup g E D. sup E does not exist

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: Suppose that A₁, ..., An are bounded subsets of R. Prove that A₁ U… U An is bounded and that sup(A₁…

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Q: 60. Let A be a bounded subset of R. Which of these statements is/are not true about A? II: Vx € A,…

A: Option (B) is correct.

Q: 1. Let C be a convex set. Let 3 0 be a fixed number and define BC = {z | z = Bu for some u € C}…

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Q: 3. In each case suppose R" has the norm ||l inclusions: = max {ril,., r.}. Prove each of the set (a)…

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Q: Exercise 2.15. Construct uncountable sets A and B such that A\ B is: (A) finite (B) countably…

A: See the attachment

Q: 1. Let Ω = (0, ∞). Let A = (3, 8] and B = (5, 10]. Find the smallest collection of subsets of Ω that…

A: Definition: Let Ω be a non-empty set. Let Σ be a collection of subsets of Ω. The collection Σ is…

Q: 5. Let R and S be relations on a set A. Use proof by contradiction to show that if R and S are…

A:

Q: 7. Show that if S x T is a closed subset of R?, then S and T are closed subsets of R.

A:

Q: 9. The set {0} in the indiscrete space X is A.open but not closed B.closed but not open C.both open…

A: In the indiscrete topology  open sets are empty set and set x itself. Hence the closed set are empty…

Q: Let S (1,2,3, 4,5,6,7,8,9, 10), G-Su, and T {1,7)CS. (a) Find G which leaves T setwise invariant,…

A: Please check the next steps for detailed answer

Q: If E <R, then for any 0 <E or an open set there exists b such that m * (G – E) < ɛ. Give an example…

A: For any non measurable set E, the above relation does not hold. The Example of such non measurable…

Q: Show that if A is closed in Y and Y is closed in X, then A is closed in X.

A:

Q: Let A be a compact set in R. Show that the complement A is open and contains an interval of the form…

A: Solution: Before going into the problem we must know the Theoem named Heine-Borel Theorem.…

Q: on as 3. Let A and B be bounded subsets of IR such that for each a₂ EA, there exists a b1 EB…

A:

Q: 10. Prove that for subsets X, Y, and Z of a given universal set U (a) X-Y=X-(XnY)= (XUY) - Y (b)…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

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

A:

Q: 5. Are the following true or false, and prove your answers: (a) If A is compact and B is bounded,…

A:

Q: Show that if any subset E of H is convex then x+E is also convex.

A: The objective of the question is to prove that if a subset E of a set H is convex, then the set…

Q: Given x+ y +z= 5 and Po(2,2,1), find the following. (a) Find the equation for the tangent plane at…

A: Here is the solution

Q: 6. Let UC Rm be an open set. If h: U Rm is a homeomorphism and uniformly continuous on U, show that…

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Q: Exercise 4. A subset A of a topological space X is called dense if cl A = X. The set Q of rational…

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Q: (A,7) be the topological space defined in question 2. Let A = {1,2,3}, then the set of limit points…

A: Solution: (4) Answer: The set is  A=1,2,3 Clearly, set A is finite.  We know that a finite set does…

Q: 6. Let's say the set X= {1, 2, 3, 4 is given as the whole space and M = {Ø,{3},{4},{1, 2},{3, 4},…

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Q: 3. Let X be a topological space. Show that for every subset A of X and for ever U we have AQU =…

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Q: If M and N are two norms on space X then prove that M and N are equivalent if and only if a subset…

A: Any two norms M and N on a norm space is called equivalent if  M≤N≤M. in the given question we have…

Q: 2. Determine the partition of B = {1,2,3,4,5,6,7,8,9,10} induced by the equivalence relation B…

A:

Q: Ex 4.) Let Sbe a non empty bounded set in 1. aj Let azo, and let a S:= {as: SES}. Prove that inf…

A: Let S be any -empty bounded set in a) Let  , and let  Since,   is a bounded subset of , Sup and Inf…

Q: 6. (a) Let A, B, be subsets of R. State the conditions which make the pair (A, B) Dedekind cut. (b)…

A: It is given that Let A, B be subsets of R.

Q: 4. Let Y + 0 and let A be a collection of subsets of Y satisfying the following two properties: (a)…

A:

Q: 7. Let A be a bounded subset of R and let sup A = B. Which of the following is/are not an attribute…

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Q: 4. Assume that S, T C R are two non-empty bounded sets. Prove that: (i) SUT is a bounded set; (ii)…

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Q: 1. Prove that for all sets A and B and n € Z, n ≥ 1, the following holds: n (a) U(A₁\B): i=1 = n C…

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Q: 5.3.8 Prove that if A CR is a finite set, then A is well-ordered.

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Q: If the point A (4,3) and B(x,5) are on the circle with centre 0 (2,3). Find the Value of x

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Q: 4. Determine which of the following subsets of R2 are open, closed or none of them. (a) {(x, y) =…

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Q: Assume CCR is closed and a € C. Set B = C \ {a}. When is B closed? (Find necessary and sufficient…

A: C⊆R is closed and a∈C.Set B=C \ a.

Q: . Show that every map f : A → B of sets decomposes as hog where g : A → C is surjective and h : C →…

A:

Q: 1. Let M, d be a metric space with A C B C M. Suppose that B is totally bounded. Prove that A is…

A: Let (M, d) be a metric space and let A be a nonempty subset of M. The diameter of A is defined as…

Q: 4. Let X be an ordered set, z E X, and SC X. Prove that x = sup S, if and only if Vy EX (yy).

A: