For any collection C of subsets of a set X there exists a smallest algebra S of sets which contains C (i.e. there exists an algebra $ containing C which is such that if F is any algebra containing C, then F contains $.

Q: Prove for all sets of A, B, that P(A) U P(B) is a subset of P(AUB).

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Q: Let Y be a G-set. Show that the action of G on Y induces a homomorphism Þ: G → Y

A: Given a group G acting on a set Y, we want to show that the action of G on Y induces a homomorphism…

Q: Consider the relation R on the set Z defined by a Rb if |a| = |b|. Determine if R is reflexive,…

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Q: For any collection a of subsets of X, there i ntaining a.

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Q: 8. Let X = {1,2,..., 15}, A = {1, 2, 3, 4, 5, 6, 7}, B = {5, 6, 7, 8, 9}, and C = {2, 3, 6, 8, 10,…

A: Please find the attachment I use the definition of sigma algebra

Q: Prove that intersection of any class of convex subsets of N is convex?

A: Let's suppose that A and B are 2 convex sets. We have to show that A∩B is also convex. Let's take…

Q: set X. Determine the of 1 x generated by

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Q: Recall that sets A and B are called equivalent (to each other) provided that there exists a…

A: Two sets 'A' and 'B' are similar/ equivalent/ equipotential if and only if there exists a bijection…

Q: Let X = {a, b, c, d, e}. Create a σ-algebra X from set X. Explain why it is a σ-algebra.

A: Given : The given set is  X = {a, b, c, d, e} To find : We have to create a σ-algebra from the set…

Q: Let X = {1,2,..., 15}, A = {1, 2, 3, 4, 5, 6, 7}, B = {5, 6, 7, 8, 9), and C = {2, 3, 6, 8, 10, 11,…

A: Given: X=1,2,3, ... , 15 A=1,2,3,4,5,6,7 B=5,6,7,8,9 C=2,3,6,8,10,11,12 To find: a) σA, B, C b)…

Q: Equation: Answer:

A: We want to find value of x. For this we use Pythagoras theorem, which states that Hypotenuse2 =…

Q: The class of all finite disjoint unions of intervals of the form (a, b), (-0o, b), (b, o0), 0, and R…

A: Given: The class of all finite disjoint union of intervals of the form (a,b] ,(-∞,b] ,(b ,∞) ,∅ and…

Q: Prove that empty set is subset of all sets: φ ⊆ A

A: Given: An empty set and set A is given. To Find: To prove that empty set is subset of all sets: ϕ⊆A

Q: Say a set A is countable if there is an injective map f: A → N. Show that the product of countable…

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Q: Prove that G is open in a space X if and only if Gnà = GNA for every subset A of X. topology problem

A: Let G be open in a space X and A be any arbitrary subset of X To prove G∩A¯¯=G∩A¯ For any subset A…

Q: S= {(A, B): (A, B e P(U))^(A c B)} %3D

A: Given : Prove or disprove each of the properties of reflexivity, symmetry and transitivity of S. Let…

Q: Show that the subset A of cantor set is not homeomorphic to A+A and A*A

A: Since, A is a subset of counter set ∩nFn and A1=A+A is also a subset of ∩nFn then, A2=A*A⊆∩nFn Case…

Q: Let A be the set of all permutations of the 26 letters in the English alphabet. The relation R on A…

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Q: Prove or disprove If P is ordered set and f is an order preserving then P is isomorphic to f(P)

A: The statement "If P is an ordered set and f is an order-preserving function, then P is isomorphic to…

Q: Let A be a finite set, and suppose that f : A → A is one-to-one. Does f necessarily have to be onto?…

A: We are given a finite set named "A" and we are given a one-to-one function from A to A. We need to…

Q: Suppose f, g, and h are all mappings of a set A into itself. a. Prove that if g is onto andf 0 g = h…

A: Given that, f,g and h are all mappings of a set A into itself. If g is onto and f∘g=h∘g  To prove…

Q: Prove that for any sets A, B there is an equivalence A × B ~ B × A.

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Q: We have G be a o-algebra of subsets of a countable set Y such that for every distinct x, y € Y there…

A: We know, If G is σ-algebra , then by definition of σ-algebra ϕ∈G If A∈G then  AC∈G Countable union…

Q: et S ⊆ N, and for any a, b ∈ N, consider a relation R such that aRb if and only if there exists c ∈…

A: This is the problem of Relation.

Q: Prove or disprove: for any set A, there exists a relation R on A such that R is both symmetric and…

A: Let  A be any we define a relation R on A s.t

Q: Let X be a set and {Ya|a E A} an indexed family of sets. Prove that X U (Naea Ya) =Naea (X U Ya).

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Q: Let R1 and R2 be two different relations on the same set A. If R1 ∩ R2 is symmetric then R1 and R2…

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Q: Let F be a collection of subsets of a set X. Then the intersection A of all o-algebras of subsets of…

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Q: Let X = {1,2, 3, 4, 5, 6}, A = {0, X, {1,2, 3}, {3,4}, ...} is a o-algebra on X. Which of the…

A: Given set is X={1,2,3,4,5,6}

Q: Either provide an informal proof of OR a counterexample to each of the following. If g and h are…

A: If g and h are relations on R x R, and g and g ∘ h are both one-to-one, then h is one-to-one. When…

Q: Let X be a set. (i) Show that the set of all subsets A of X such that A is finite or X\A is finite…

A: Given Information:i) Set of all subsets  of  such that either  is finite or  is finite.ii) Set of…