Let A, B, and C are three sets, prove that Bx(AUC)=(B× A)U(B×C)

Q: (b) Consider the set R. Prove: (i) VxER, x 0 = 0; (ii) If x y = 0 then either x = 0 or y = 0, or…

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Q: Let u= and v= Show that is in Span (u, v} for all x and y.

A: Given u=4-1 , v=41 and b=xy. The vector b=xy is in span u,v if the system containing u,v and b is…

Q: Let (X,T) be a toplogical space, A, BC X. Show that АС В Ас в, АсВ. 介

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Q: Complete the proof: If A≤B and B≤C, then ACCn(transitive property for set in inclusion) Need to…

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Q: Prove that (C, +) ≈ (R × R, +).

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Q: 6. Sketch the set X = {(x, y) E R? : y<x²} on R2. Shade in the set X. %3D

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Q: A is symmetric

A: In this question, we have to check the matrix A is symmetric or not.

Q: The projection of u onto v is given by projvu = ________.

A: The projection of u onto v is given by projvu = compvu.

Q: A ° = A – JA

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Q: Assume X, Y CU. Prove that (U \ X) U (U \ Y) = U \ (X n Y)

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Q: Give an example of a compact set A and a closed set B in R2 such that (conv A) N (conv B) = Ø but A…

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Q: Let X and Y be two T, spaces, then X×Y. :is

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Q: 4).ェ Pare subset, -f y such that CeD than fic) is not eD, then the Convers %3D そイue.

A: Solution As it's mention C=D and f^-1 same , so each and every element from  f inverse we get equal.…

Q: 1. Prove that for all sets X, Y , if XU Y = Xn Y, then X = Y. %3D

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Q: If CI(A) denotes the closure of A, which of the following is not correct in general for a set A: *…

A: cl(A) denotes closure of A

Q: Let R have its usual topology and let X = {a,b, c}. Define f : R –→ X by а, х

A: Given, ℝ have its usual topology and let X=a, b, c. Define f: ℝ→X   by…

Q: Show that if A is closed in X and B is closed in Y then A × B is closed in X × Y

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Q: Suppose (X,S,u) is a measure space and f,h: X → F are S-measurable Prove that I|fh||, < ||f||, ||4|a…

A: Given X,S,μ is a measure space and the functions f,h:X→F are S-measurable. Suppose 1p+1q=1r,…

Q: Let A=(0, π ) υ (π e?] %3D c. Is A a closed set? d. Does A contain any isolated points? е. Find the…

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Q: Let (X, T) be a topological space and A C X. Prove that A =A.

A: PROOF :

Q: Let W = Lin and U= Lin 2 (a) Show that R' = W U. Explain.

A: Note:- As per our guidelines, we can answer the first part of this problem as all the parts are…

Q: Let (X, CA) be a measurable space, f: X (-00, ws) and 1: X (-∞, ∞0) are measurable. Prove that the…

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Q: Sketch the set of points satisfying: |z - 1 + 2i| = |z + 3 + i|

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Q: lar, let m = 2 and n e of A and B such that

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Q: Exercise 2. Prove that if A = B and C z D, then A × C = B x D.

A: Given data: A≈B......(1)C≈D......(2) The objective is to prove A×C=B×D

Q: Suppose X has n elements and Y has m elements. How many bijections X → Y are there?

A: To find: The number of bijections X → Y are there. Given information: X has n elements and Y has m…

Q: Let (X, M, u) be a measure space wE

A: Given, (X, Μ, μ) be a measure space where μ is a probability measure. Assume thatf, g : X→ℝ are…

Q: Define the Cartesian product of sets X and Y? How is this Cartesian product denoted?

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Q: Let f(z, y, z) = z2 +y? + z²,z = /utv, y = /u – v, and z= uv, Find in terms of u and v.

A: we have to partial derivative with respect to u by using chain rule

Q: Prove: If U is a neighborhood of x0 and U ⊂ V, then V is a neighborhood of x0. Also, if U1 ..., Un…

A: The neighborhood of a point: Let (X, T) be a topological space and x0∈X. A neighborhood of x0 at is…

Q: Let V C(R ) be a nonempty open set satisfying a – b EV whenever a, b e V. Prove that V = R. %D

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Q: Let X be an uncountable set and let T = {U C X :U = Ø or U¢ is finite} %3D Then is topological space…

A: let X be an uncountable set and T=U⊑X : U=∅ or Uc is finite i.e cofinite topology on X. Then the…

Q: show that d connot be obtained from a norm.

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Q: If (a + bx)e = x, prove that x’y, =(xy,-y)

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

Q: Prove A = BH (x)A = (Vx)B without using the equivalence theorem.

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Q: O Consider the set R. Prove: (i) VxER, x 0 = 0; (ii) If x y = 0 then either x = 0 or y = 0, or x = y…

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Q: 3. If u= (x+y +z') , prove that +y

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Q: 3) let X be discrete topology and Ac X, find A ?

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Q: (iii) If (X,r) is topological space, A,BCX. Prove that (a) b(A)=AA (b) If ACB, then d(A) ≤d(B)

A: First part Is obvious from definition of boundary points of a set.

Q: Prove that u, v, and w are all in span(u, u + v, u + v + w).

A: To Determine: prove that u ,v and w are all in span (u,u+v,u+v+w). Given:  we have(u,u+v,u+v+w)…

Q: Let X = R, and M = {0}, then span (M) = %3D %3D OR O {0}

A: If, M = {0} then, span(M) = {0}

Q: Q1/ Prove that (Sa,0) is not Comm Sroup.

A: To prove the non commutativeness of the group

Q: • Show that if A is compact and Bis closed, and d(A, B) = 0, then ANB÷Ø.

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Q: Show That R is Banach Sface.

A: To Show -

Q: (b) Let (X, >, ) be a measure space. For A, Be ). Show that μ(AU B ) - μ(Α) + μ(Β) -μ(An ). %3D

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

Q: . Prove with the usual notations that: (b) A +V =- (c) A= SEŽ (a) EV = VE = A

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

Q: Let 9-(a: a€R}, then T is Borel algebra.

A: Clearly the measure of the set is =0 .

Q: Sketch the set of all points (x, y) such that x + y| < e*.

A: Consider the function x+y≤ex. An absolute function is defined as x=x if x≥0-x if x&lt;0. So, there…

Q: Let X = {a, b, c, d, e} and let T ={X, Ø, {a}, {c,d}, {a,c,d), {b,c,d,e}), therm %3D O (X,T) is…

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Q: Assume f,g : [a, b] → [c, d] where a < b and c < d. Prove that sup{f(x)+g(x) : x € [a, b]} <…

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Statements and Logic

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