Is there a set A that satisfies A = {A}? If yes, exhibit one such. If not, Why not exactly

Q: Prove that a subset of a set of measure zero has measure zero.

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: Define a smallest possible set of which both {1,3,5} and {0, 3,4 } are subsets.

A: The subsets given are {1, 3, 5} and {0, 3, 4}.

Q: In the discrete topology on the set X, every set is open and it also follows that every set is…

A:

Q: Let A be a set and let B be a proper subset of A. Is it possible for B to have the same cardinality…

A: Given: Let A be a set.And B be a proper subset of 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: Prove that the smallest member of a set, if exits, is the greatest lower bound of the set.

A:

Q: Assume that the set S has 14 elements. How many subsets of S have at most 4 elements

A: At most 4 means 4 or less.  Let us denote X=number of elements

Q: Give an оп ехат each of which example of a set with three elements is a xet.

A:

Q: Suppose a set A is uncountably infinite, that U is countably infinite, and that UC A. Show that A -…

A:

Q: Let A be a denumer able set. Prove that A has a denumer able subset B such that A - B is…

A: Let A =set of multiples of 4, B=set of multiples of 3. Both are denumerable. Intersection of A and B…

Q: Show that every open set with respect to 7 is a union of open ball(s).

A:

Q: Let U be universal set of all the students of Class X of a coeducational school and A be the set of…

A: In a coeducational school, there can be only boys and girls in school. Since, A is the set of all…

Q: Give an Such that. example of sets A andB

A: Universal Set: We consider all the sets to be the subsets of a given set known as the universal set…

Q: 2. Is there a set A that satisfies A = {A}? If yes, exhibit one %3D such. If not, Why not exactly?

A: No, A≠{A} for any set A. Let  A=1,2,3 then number of elements in A are 3. Now,  A=1,2,3 here the…

Q: Prove that (2,5) is an open set

A:

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: Show that the o-algebra B of Borel sets contains all F,-sets.

A:

Q: Is the Cantor set closed? Is it compact? Explain!

A:

Q: Suppose that X and Y are sets such that the power set of Y is a subset of the power set of X. Does…

A:

Q: Show that any convex open set is simply connected.

A: The objective is to prove that any convex open set is simply connected. Simply connected is nothing…

Q: Assume that the set S has 13 elements. How many subsets of S have at most 4 elements?

A:

Q: Prove that the Cantor set has measure zero.

A:

Q: subset

A:

Q: Prove that the greatest member of a set, if exits, is the least upper bound of the set.

A:

Q: Is there a set A that satisfies A = {A}? If yes, exhibit one such. If not, Why not ?

A:

Q: Prove that every infinite set S contains a proper subset similar to S.

A: We have to prove that every infinite set S contains a proper subset similar to S. Let X=S Let X be…

Q: prove that empty set is a subset of every set

A: We have to prove that empty set is a subset of every set = ?

Q: Prove that a set is closed if and only if its complement is open

A: prove of the given problem is given below...

Q: If A is a countable set and B is an uncountable set, then the MOST we could ever say about A U Bis…

A: We have given the three MCQ questions about countability and uncountability . We have to choose the…

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

A: Given problem is :

Q: Consider the set A = {xy : x ∈ N, y ∈ N} = {1, 2, 22, 23, 24, ...., 3, 32, 33, 34,......, 4, 42, 43,…

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: open

A:

Q: Prove that if A is a set then so is {A} but do NOT use an argument that involves stages explicitly.

A: 3 Given A is any set. We have to prove that A is also a set. Since, A is any set Then as every set…

Q: Let A be a well-ordered set, then every subset of A is not well-ordered

A:

Q: prove that the union of a countable set and an uncountable set is uncountable

A:

Q: 3. Prove that if A is a set then so is {A} but do NOT use an argument that involves stages…

A: Let A is any set then as every set is a subset of itself then, A⊆A⇒A∈PA⇒A∈PPA

Q: open ball is the open set

A:

Q: Let A1,A2,.......,An be countable sets. Prove that  A1 x A2 x.............An = { a1, a2,…

A: Let A1,A2,.......,An be countable sets. Then we have to show that    A1 x A2 x.............An = {…

Q: Prove that every set with positive Lebesgue measure contain a non-measurable set

A:

Q: Let A and B be subsets of a universal set U, show that ACBif and only if BCA.

A:

Q: B. Prove that the union of a finite number of compact subsets F1, F2, ..., F, of RP is a compact…

A:

Q: Prove that the set of all finite subsets of ω is countable. (Set theory)

A: The objective is to prove that the set of all finite subsets of ω is countable.

Q: Prove that the union of any number of open sets in Ris open.

A:

Q: Let A1, · · · , An be countably infinite sets. Prove that A1 × A2 × · · · × An is countably…

A:

Q: Consider the set X={a,b,c,d} construct all the possible topologies of X that have only three open…

A: Given: The set X=a,b,c,d To find:  all the possible topologies of X that has only three open…

Q: Show that the union of two sets, each of measure zero, has measure zero

A: