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

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

A:

Q: Give an example of each of the following or explain why you think such a set could not exist. (i) A…

A: (i). Such a set could not exist. Let  be a nonempty subset of a topological space . Let . If  is not…

Q: Define Open set ?

A:

Q: Prove or disprove: For all sets A and B, An(AUB)

A: We have to prove or disprove , for all sets A and B A∩(A∪B)=A.

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

A: No, there is no set A that satisfies A={A}. Let A is an empty set, i.e.  A=  or  A=ϕ, in which…

Q: Prove that the smallest member of a set, if exits, is the greatest lower bound of the 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: Find an example of nonempty, distinct sets A, B, C for which (ANB) UC = (AUB) nC.

A:

Q: The continuous image of a connected set is connected. That is, if X is connected and f: X→ Y is…

A:

Q: 2- Prove that H, is convex set.

A: Let us look at an arbitrary closed half-space defined as H+={x∈ℝn:aTx-b≥0}, representing a half…

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

A:

Q: Give example open set is compact

A:

Q: Choose any that is a subset of the set (Ø} {{Ø),0} {{1}}

A:

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

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: What is meant by the Cardinality of a finite set? Give one example.

A: The cardinality of a set is meant by the number of elements of the given set. If the set has no…

Q: Prove that if S is a bounded set then S is bounded.

A:

Q: Use the pick-a-point method to show that if A and B are subsets of some universal set U, then A ∪ (B…

A:

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

A:

Q: Show that if A is an infinite set, then whenever B is a set, AUB is also an infinite set.

A: we have to prove that AUB is infinite set.  Since, given A is infinite set. since we know that…

Q: Use the method of intersection of sets to compute the following. (a) GCD(12,16,8) and LCM(12,16,8)…

A:

Q: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

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: Use the definition of compact set to prove that the set [0,1] is compact

A: We know that a set E is compact if every open cover of E has a finite subcover. We suppose, that O…

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: two example of open set in R, closed set in R, and the closure of that set

A:

Q: Let X be an infinite set, then any topology on X is also infinite. Select one: O True O False

A: Note: " Since you have asked multiple question. As per our guidelines we are supposed to solve only…

Q: Let A be a finit set and let |p(A)|=n. Let B = A U {x} for some x∉B, How many subsets does B have…

A: Let A be a finite set with m elements Then number of elements in p(A) has 2melements Given p(A) =…

Q: List all the possible subsets of set G, where G = {-7,m,z}- %3D

A:

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

A: Set equality

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: Define a set of which both {1, 3, 4} and {0, 3, 5} are subsets. Find such a set with the smallest…

A: {0,1,3,4,5} is the smallest set that contains {1, 3, 4} and {0, 3, 5} as sub-sets.

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

A: Given problem is :

Q: Show that any subset of bounded set is bounded

A:

Q: Give a set that is not closed under addition

A: In the question we have to give a set that is not closed under addition.

Q: let A Set bounded belou, erove that C. inf A Inf C A ン

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: Give an example of a set in R ^ 2 with the usual topology that does not have any subset (not…

A: Given that R2 with the usual topology. The aim is to find an example of the set  R2 with the usual…

Q: Show that if A is an infinite set, then whenever B is a set, A ∪ B is also an infinite set.

A:

Q: ndary of a path-connected set in R^2 with the usual

A: We have to check that boundary of a path connected set in R2 with the usual topology is…

Q: 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…

A: Let A be a non-empty set and Cii∈I and Djj∈J are two different covers of A. Since Cii∈I is cover of…

Q: Prove that a set A is connected iff A is and interval.

A:

Q: Prove that if A is a transitive set then UA is a transitive set.

A: If A is a transitive set then, It is proved that ∪A is a transitive set as shown below.

Q: Ifs is(finite or infinite) Contable Set then prove that M*(S) ==

A:

Q: 2- Moke the definitions of open and closed sets. x =Larbic,d,eļ and ノ closed subsets of (x,T) space,…

A:

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

A:

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.