T have to show rigorous proof of the inclus (An B) x C c (Ax C)UCBXC) 1,B.and C, Sets non-em pty of acniversal set U.

Q: Prove that for any sets A, B, C: (AUB)-B A - B

A:

Q: Show that relint (C) is nonempty for any nonempty convex set CC Rª.

A: Relative interior of C : Let C be a convex set. We define the relative interior of C as the set of…

Q: 4: Ex 4.) Let S be a non empty bounded set in R. a ( a. Let azo, and let oS := {as: SES}. Prove that…

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Q: When is a set compact?

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Q: Without using the fact that the complement of an open set is a closed set, prove that if SCR such…

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Q: |x²| = p^ n

A: Given: X=pWe need to show that Xn=pn

Q: Question: 7. Suppose A and B are sets in universe U. Prove that An (AUB) = B\ A.

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Q: a) A = B = R and f+g: R → R is defined by (f + g)(x) = f(x) + g(x); b) A = B = R and fg: R → R is…

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Q: 1. Determine which of the following statements are true for all sets A, B, C, and D. If a double…

A: .

Q: QLet (X,t) be a topological space and A,E are two subsets of x show that : 3. A is closed set iff A…

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Q: Q.Prove If ANB is pathwise connected set, then A andB are or disprove the following: pathwise…

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Q: tor all l sets A, Bad C やok that A xCB-c) = (AX1B J- CAxc). %3D

A: Given below the detailed solution

Q: 1) Let the set S be defined as S = {x | x is a U.S. President} Find n(S). 2) Define compliment to a…

A: use the basic property of sets .

Q: (b) Let = {A; : i EN} be an indexed family of sets with the property that A₁ A₂ 22 A, 2..., that is,…

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Q: [Falkner Section 11 Exercise 15(b) – modified] Let S and T be sets. Prove that S and T are disjoint…

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Q: Prove that: (i) A is closed. (ii) A = nc. ก ACC closed (iii) A is the smallest closed set in X that…

A: Given the set A which is the closure of the set A, i.e. a set that contains all point of A as well…

Q: Let A and B be sets. Prove that Ac B if and only if AnB = A.

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Q: b) By using the laws of set theory, show that [(A U B) n (An C)] u (B n C) = A U B.

A: Here we have to prove left hand side is equivalent to right hand side using laws of set theory

Q: Jive R² which are example of two subsets homeomorphic. not C specify the set explicitly) g

A:

Q: Describe the set {TER²: doo (r,(0,0)) = 1}.

A: Given: Set x∈R2:d∞(x,(0,0))=1 We have to describe the set.

Q: Assume all sets are subsets of a universal set U. Suppose that A, B, and C are arbitrary sets, Prove…

A: Given, A-C∪B-C=A-Bc∪C

Q: prove or dis Prove a Every convex set of avectory is affine set- • Every affine set is sub space

A: A convex set is a subset of a vector space (such as Euclidean space) where for any two points within…

Q: X, Y and Z be subsets of the set . Verify the following properties: XCY XUY=Y⇒XnY=X -UI 3)

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Q: If A, B, C are sets, show that a) A x B= 0 ⇒ A=0 or B = 0. b) (AUB) x C = (AXC) U (BxX C). c) (An B)…

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Q: 1) Let f: A ->B with C1 C C2C A and Dic D,CB Which is true? a) f(C) c f(C2) OR b) f(C2) c f(C:) (i)…

A: Given: i) To prove that answer is correct. ii) To give a counterexample to show that the set…

Q: a) veriry one of DeMorgan's Laws for sets! {S\U: U €U} = S \ U{U: U €U}. b) Show that the…

A: This is a question of topology.

Q: Prove that Everry subset of negigible seti's gigible

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Q: 3) Let S be o st defined os follow s: 05 Deter mire ue ther S is o con vea set or not.

A: We have,  The set C is said to be convex set if for any u,v∈C, the point tu+1-tv∈C for all t∈0,1…

Q: sense, give rea somable membership common function for the followng fuzzy sets. (of course must…

A: a)- Given: x is much larger than y. Let z be a function. So, the set is: z=x-y,     x, y∈ℝx-y>0

Q: show that empty set, X is an element of S intersection T in topology

A: Here We have to show that empty set must be in S∩T where S and T are topology on set X.

Q: Suppose that U1, U2, ..., Un is a collection of n open sets. Prove that their intersection…

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Q: Let X be a nonempty set with subsets, A, B, and C. Prove that A ∩ (B \ C) = (A ∩ B) \ (A ∩ B).

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Q: Decide if each of the following sets is open, closed, compact and find its closure: b) Qn [0, 1] c)…

A: b) Consider the set: Q∩0,1. Consider: x=12. It is an irrational number that lies between 0 and 1.…