Prove that the intersection of any number of closed sets in A is closed. 6.27 Prove that P is a limit point of a set S in E if and only if every open set co point of S other than P. 6.28

Q: 7. (10 points) Let f: A B be onto. Prove that S((C)) = C for all sets CC B.

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Q: Show that the function h: AUB → {1,..., m+n} in the proof of the first counting principle, defined…

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Q: Prove that there is no set to which every se Show that for any nonempty set A, there xE Bex belongs…

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

Q: 7. Let x E R and A be a nonempty subset of R. If x is not a limit point of A then x is an isolated…

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Q: 1. (p.209, #16) Define a relation R on Z by declaring that xRy if and only if x? = y? (mod 4). Prove…

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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: 2.28 let {Ei} i Ej be a Measurable pairwise dis joint sets. prove that for any bounded set A…

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Q: 2. Find a limit point of the following set S. Is it a closed set? Why? S = { 1 nez.}. n +1

A: We use some results of limit point of set. And use the theorem of closed set.

Q: Define the relation ~ on the set Z of integers by m~n if and only if m^2 = n. Show that ~ is not…

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Q: Recall that for any set SCR we denote the boundary of S (i.e. the set of all boundary points of S)…

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Q: 5. Prove: Let A₁, A2, be measureable sets such that limn→∞ µ(An) Im C = 0. Then for every

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Q: Let S={a,b,c,d,e,f,g} be ordered as in the following Fig. f, e C a b And let X={c,d,e} The upper…

A: Suppose B is a subset of partially ordered set A. An element m∈A is said to be upper bound of B if…

Q: is not an endpoint of any of the intervals (e) Prove that eC but in any of the sets A (k e N),

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Q: of points of S. 6.30 Show that a set consisting of a finite number of points in E is bounded. Gnita…

A: Given, a set consisting of a finite number of points in E.

Q: 4. Suppose Rand S are symmetric relations on a set A. Prove that RoS is symmetric iff Ro S = So R.…

A: Suppose R∘S is symmetric Let s, r∈S∘R ⇔r, s∈R∘S⇔s, r∈R∘S ( since R∘S is symmetric)  So, R∘S=S∘R

Q: 2) Let A be a finite set and a EA.Lf a EA is not a maximal element of A, then a has an îmmediate SUC…

A: Given A is a finite set. a∈A is not a maximal element of A. Now we have to prove 'a' has an…

Q: Given a relation R on Z defined by mRn ⇔ 3|(m −n) for all m, n ∈ Z. Find the equivalent class…

A: We have the relation defined by mRn⇔3/m-n Now ,we need to find the equivalence class determined by…

Q: If X #0, compare the following topologies on X: a) The co-finite topology 7, and the co-countable…

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Q: 9. (i) Prove by mathematical induction that for n e N, Ar? – 1 2n + 1 (ii) A relation R, is defined…

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Q: 11. E is the relation defined on Z as follows: for all m,n e Z, mEn → 4|(m-n). a) Prove that E is…

A: Given a relation E on Z as follows: for all m,n∈Z, mEn⇔4m-n.

Q: 2. Let E C R. Prove K is compact if and only if every infinite subset of K contains a limit point of…

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Q: Show that every open set with respect to 7 is a union of open ball(s). A point p is a limit point of…

A: Given X be a matric space with metric d. For any r&gt;0, the open ball at center at x∈X  with radius…

Q: 5. Show that the set {e" : n e Z} is countable. (Hint: Use the function f(n) = en and show it is 1-1…

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Q: 4. Let Š and T be subsets of R. Prove the following: a] ci(c!S)= clS

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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: 2.6.4 Let A CR be a non – empty set. Suppose that A is bounded above. Let U = {x E R|x is an upper…

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Q: b) Consider Hasse diagram for a poset, P as given below. Find the following, if exist(s): (i) the…

A: In this question we have solve the Hesse diagram for the minimal and maximal elements, greatest and…

Q: Let S be the set of integers. If a, b E S, define aRb if ab &gt;= 0. Is R anequivalence relation on…

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Q: b( Let S be the relation defined on P ((1, 2, 3, 4}) defined by X SY if and only if IX| = |Y| (mod…

A: The complete solutions are given below

Q: 2.12 let A and B bounded sets For which there is an ε >o such that la-blze for all ac A, be B .…

A: Consider A and B are bounded sets.

Q: Let f : X → Y, A¡ C X for all i e I and B; C Y for all j e J where I and J are some nonempty index…

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Q: Let f : X → Y, A¡ C X for all i e I and B; C Y for all j e J where I and J are some nonempty index…

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Q: 9.C. Prove directly that if K is compact in Rº and F is a closed subset of K, then F is compact in…

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Q: (7) First define a semi-finite mea sure. Then prove that if u is a semi-finite measure and (E)- then…

A: The method is given below with proper explanation and definition: Let (X,Σ,μ) be a measure space.…

Q: [2.13] Let S be a nonempty set in R". Show that S is convex if and only if for each integer k > 2,…

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Q: (2.1) Let b> 0 and X [-b, b). Here we admit %3D B"open" in r if and only if (0} B or (-b, b) C B.…

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Q: Consider the equivalence relation E on N defined by Enm iff n-m is divisible by 4. Which statement…

A: An equivalence relation E is defined on the set of natural numbers N as  Enm denotes that n is…

Q: Let D = {2,4,6} and E = (1,3,5}, define a relation R from D to E as follows, given that…

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Q: 5. Suppose E is a given set, and On is the open set: On = {x : d(x, E) < 1/n}. Show: (a) If E is…

A: Given that E is a set, and On is the open set: On=x:dx,E&lt;1/n To prove (a) and (b):

Q: Consider the set Z in which the relation R is defined by aRb if and only if a + 3b is divisible by…

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

Q: of E. 4. Let E CR be a bounded set. Prove sup E e Ë.

A: (4) Let E be a bounded subset of R  and let u be the sup E.  Then two cases arise-  Either u belongs…

Q: (a). Find a set S and a bijection between S and a proper subset of S. (b). What condition on S…

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Q: a) Suppose that E is a set. If there exists a function f from E onto N, then E is at most countable.

A: (a) False f:E → INLet E= IR  Thenf: IR → INdefened as f(x) =  1  , ∨ x ∈ IRi.e There exist a…

Q: Consider the subset G {r• /2: x € Z} of R. Show that G is closed un- der addition.

A: The given subset is: G= x·2 : x∈ℤ

Q: 5. Let S be a nonempty subset of R that is bounded above, with upper bound b. Prove that b = sup(S)…

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Q: . A finitely additive measure u is a measure iff it is continuous from below as in meorem 1.8c. If…

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Q: 2) Take any r > 0 and any c ∈ R. Show that the  r-ball Br ( c ) is an open subset of R.

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Q: 5. Without using the U R representation of R*, Prove that for any set relation R (we did not pick it…

A: We need to Prove that for any set relation R (we did not pick it to be "on" any particular set R+ =…

Q: 4. Let E be a measurable subset of [0, 1) and E+y for y E [0, 1) be a translation modulo 1 of E.…

A: Given that E is a measurable set. Consider the set E+∘y=x+∘y : x∈E where x+∘y is defined as…