1.5 Prove that all the a-cuts of any fuzzy set A defined on R" are convex if and only if such that r, s EN", λ E [02]

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: Prove that the map g : Z →→ N defined by | 2k, -(2k+1), k < 0 k 2 0 f (t) = is one to-one nd mans…

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Q: 1. Prove by Induction that the numbr f edges of Kn

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

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Q: Prove that the set S = {n² : n e N} is denumerable. %3D

A: We have to show that the given set S is a denumerable set.

Q: 1.4 Determine whether the following fuzzy sets are convex or not. a) A- Ju,(x)/x where ,(x) =1(1 +…

A: (a) A=∫μA(x)/x  where μA(x)=11+x2, -∞&lt;x&lt;∞ It is clear that z=λx+(1-λ)y, 0≤λ≤1 For x,y ∈ℝ…

Q: Prove that 10. Let f(0) = 20 be the map of S periodic points of f %3D are dense in S.

A: From the set of reals R, choose a subset P such that each and every point of R lies in P or if you…

Q: Theorem 6.2. Let C be a compact subset of Rstd· Then C has a maximum point, that is, there is a…

A: We start the proof by assuming there are no such maximum element in C.

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

A: Since S ⊆ T it follows taht inf T ≤ t, ∀ t ∈ T, then  inf T ≤s , ∀ s ∈S:Therefore inf T is  a lower…

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: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

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Q: Theorem 6.2. Let C be a compact subset of Rstd- Then C has a maximum point, that is, there is a…

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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: Exercise 6.40. Let E be a nonempty subset of R that is bounded above. Prove that if a = sup E, then…

A: This is the question of topology.

Q: Q3. Let N = {1, 2, 3, ..} equipped with the cofinite topology. Prove that M = {7,14, 21} is a closed…

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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: Let M and M2 be two R-modules and N, and N2 be two submodules of M, and M2 respectively. Also, let…

A: We will use fundamental theorem of homorphism here . If f : G--&gt;G' is onto homorphism then G/kerf…

Q: 7. Let X be a set with |X| = n € Z4. Prove that for any set Y, |Y| = n if and only if there exists a…

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Q: 2.1 Let B be the set of all bounded sequences of real numbers and define the function d: B x B R by…

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Q: Let {Aq | a E I} and {B3 | BEJ} be two indexed collections with nonempty index sets I and J. Prove…

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Q: 2.6.13 LetE be a subset of a metric space X. Prove that E has empty interior (E° = Ø) if and only if…

A: Dense set and interior set

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: Let A be the subset of R defined by A := {1-Q" :n eN}. Find all cluster points of A and justify your…

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Q: 1.5 Prove that all the a-cuts of any fuzzy set A defined on R" are convex if and only if H.(àr + (1-…

A: Let us assume without loss of generality that minμA(r),μA(s)=μA(r)=α. Let us assume a-cut of fuzzy…

Q: Q3. Let N = {1, 2, 3, ...} equipped with the cofinite topology. Prove that M = {7,14, 21} is a…

A: Given below the detailed solution

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: Let (X,T) be a topological space,  K a compact subset of X and (Fn)n∈ N a family in P(X) with Fn≠ø…

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Q: 1. Consider the collection M = {0, X}. Obviously, X E M. Note that X = 0 is in M. Since M only…

A: Suppose A be any set. Then, Power set of A is defined and denoted as:P(A)={B:B is a subset of A}

Q: Theorem 6.17. Every compact subset C of R contains a maximum in the set C, i.e., there is an m E C…

A: To prove that, every compact subset C of ℝ contains a maximum in the set C, that is, there is an m∈C…

Q: Question 2. Prove that a nonempty closed subset of R, if it is bounded from below, has a least…

A: The given question is related with real analysis. We have to prove that a nonempty closed subset of…

Q: 2 Let A be a fixed nonempty subset of set X and I= {Bc X | A Z B}. Show that (X,I) %3D is a matroid.

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Q: Let (S,≼) be a linearly ordered set with the Least-Upper-Bound Property. Let A and B be non-empty…

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Q: Let M, and M2 be two R-modules and N, and N2 be two submodules of M, and M2 respectively. Also, let…

A: Given: M1 and M2 be two R-modules and N1 and N2 are submodules of M1 and M2 respectively. Also,…

Q: 3.6 Let A be a subset of a metric space (S, d). Show that if H is a nonempty open subset of S and An…

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Q: 6. If A has m members and B has n members, prove that A x B has mn members 7. Sketch each of the…

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Q: 11 Prove that if A CR and |A| > 0, then there exists a subset of A that is not Lebesgue measurable.

A: As per the policy, we are solving a first question. Please repost the question and specify which…

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: . Let A = (0,1) and 7 = {(¹, ¹), ()¹₂)...}. Show that is open cover for A Furthermore, determine…

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Q: *7. Let S be a nonempty bounded subset of R and let ke R. Define kS = {ks : se S}. Prove the…

A: Note: We'll answer the first question since the exact one wasn't specified. Please submit a new…

Q: 3. In each of the following examples, determine (with careful arguments) the bound- ary of A in X,…

A: Boundary of a set: The boundary of a set A in a metric space X is a intersection of A¯ and its…

Q: 1. Let X be any nonempty set and let p E X, Prove that : E, {G X:p¢ G}U{X} 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: 3.7.5. Given a finite-dimensional normed space X, prove the following state- ments. (a) X is…

A: Multipart :    Let X be finite dimensional normed space. (a) We need to show that X is separable. As…

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: Theorem 7-6. If the variables are uniformly bounded then the condition, lim Bn = 0, n- 00 n2 is…

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Q: 4.2 Let Fj, be a closed set for k = 1,2, ... ,n in (S, d). Show that U Fk is closed.

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.