An indexed family of sets (A,hcs is said to be a cover of a set X if and only if X CU,crA Use this idea to prove that if (Ches and (D,), are two different covers of a nonempty set A then (Gn DlneN forms a cover of A

Q: If  X is a countable set show that every topology on X is second countable.

A: To Show: If  X is a countable set show that every topology on X is second countable.

Q: Question 3  Show that if A and B are sets with A ⊆ B, then A ∩ B = B.  Full explain this question…

A: Show that if A and B are sets with A ⊆ B, then A ∩ B = A.

Q: Suppose V is a set and that n(V) = 10. How many subsets of V contain exactly 3 members?

A: Number of ways in which 3 elements of V can be clubbed to form a subset = 10C3…

Q: 4- A subset [a, b] R is compact set in: a- (R, Td), b- (R, T), c-Both (R, Tu) and (R, T₁) d- No one.…

A: As per our guidelines we are supposed to answer only first part. Kindly repost other parts in the…

Q: Cantor set is measure at 1

A:

Q: Let Gbe an open set in R". Two points x, Y e G are said to be equivalent if they can be joined by a…

A: Since every open subset of Rn can be written of countable union of disjoint open balls and each ball…

Q: 1. Let A, B abd C be sets. Prove that Au (B-C) = (AUB) - (C-A).

A:

Q: Argue that a set A is a Gδ set if and only if its complement isan Fσ set.

A: The given statement is of Baire’s Theorem.

Q: 11.0, Prove that the interior of the interin of a set is equal to the interies of that set

A:

Q: Does there exist a non-empty open set A and two elements in A that are farthest apart (among all…

A: There are no two elements that are farthest apart in a non-empty open set. Justification : Let A…

Q: An indexed family of sets (Ahess said te be a cover of a set X tand only i X CUEIA Use this idea to…

A: Let C and D be two different indexed family of sets Ci and Dj. Then C=Ci:i∈ID=Dj:j∈J Suppose C and D…

Q: Show

A: We prove this statement by using definition of Borel Set.

Q: Let U = {a,b,c,d,e,f,12,13,14,15,16,17}, X = {a,b,c,15,16,17), and Y = {b,d,f,13,15,17}. List the…

A:

Q: -prove that for any bounded Set E, There is abounded set A That is a countable intersection fopen…

A: The solution is:

Q: Theorem 1.10. A set is infinite if and only if there is an injection from the set into a proper…

A:

Q: A topological space is dis connected if and only if it contain non empty set which is both open and…

A: I give you two simple example with explanation

Q: prove that i } X is a Coun table set and P: X>Y is asurjecbive then Y is Countable

A: Note: X denotes cardinality of X, that is number of elements in the set X. If X is countable means…

Q: Let A = {Ba: a € A} be a family of sets and let G be a nonempty set. Prove Gn an (UB.) - U(GOB₂) =…

A: Given: A=Bα : α∈∆ be a family of sets and let G be a nonempty set. To prove: G∩∪α∈∆Bα=∪α∈∆G∩Bα

Q: Let R be a relation on A. Then for all n ≥ 0, prove R^n is a set. (Discrete Mathematics, powers of a…

A:

Q: If A is a set of cardinality m and B is a set of cardinality n. Prove that cartesian product A x B…

A: Introduction: Counting the number of items in a set, or its cardinality is a way to gauge its size.…

Q: Any connected component of Xis a closed set and X is disjoint Union of ts connected components,

A:

Q: 2) het A, B and non- dijoint diagranı be Three Cveyla Ony using sets von Show (B-C)Ule- A) # A-8

A: firstly, we will draw all given set region then we comparing both side

Q: Draw a ven diagram for this set (A ∩ B) ∪ (A ∩ C')

A:

Q: subset

A:

Q: For every element r in A, there is an element y in B such that (x, y) is in F. Can you please…

A: Relation: The relation from a set A to a set B is a subset of Cartesian product A×B. Function: The…

Q: ieN 5. (a) Uli,i+ 1]= (b) Nli, ieN ieN 6. (a) | Iro.i +11=

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

A:

Q: For an indexed collection of sets to have an intersection an element needs to be in every set right?…

A:

Q: Prove that every nonempty subset of a well-ordered set is well-ordered.

A: Given problem is :

Q: Suppose A, B and C are sets. Prove that A\ (BC) = (AB) U (ANC). [Prove this using elements of these…

A:

Q: Theorem 2. Let R and S be binary relations on a set X. Then, (a) (R-1)-1 = R. (b) (RUS)-1 = R-'Us-!.…

A:

Q: Is it possible for two nonempty sets to have the same intersec-tion and union? If so, give an…

A: When two sets A and B overlap that is A = B, then the intersection and union of the  sets A and B…

Q: I) We know that -5,-4,-3,-2,-1? and IN are well-ordered sets with respect to the trivial ordering.…

A: Solution: Given the sets -5,-4,-3,-2,-1 and ℕ are well-ordered with respect to the trivial ordering.…

Q: How can I show that the collection  A composed of empty set and all cofinite subsets of X Is a  sets…

A:

Q: Theorem 1.12. The union of countably many countable sets is countable.

A: We have to prove that the union of countably many countable sets is countable.

Q: There is a set X = {1, 2, 3, 4, 5}. Find the sets A, B, C such that A ⊆ X, B ⊆ X, C ⊆ X, (A ∪ B) ⊆…

A: Here we will find sets A,B,C;