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

Q: What is a set?

A: Definition of the set:

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: 4. Prove that if A and B are convex sets, then AN B is also convex.

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: Suppose that S is a non-empty finite set. Prove that ? has as many even-sized subsets as it does…

A: Let S be a non-empty set. Let A be the set of even sized subsets of S and let B be the set of odd…

Q: 4. Let S be any nonempty set with the operation a * b = a. Is (S,*) a semigroup? Is it commutative?

A: There is no such introduction for the solution.I have explained it within the solution.Please go…

Q: True or False If A is a set, the complement of A is the setof all the elements in the universal set…

A: Complement of a set A: Consider a set A. Then the complement of the set A is the set of all elements…

Q: 3.1 Show that a finite intersection of closed sets is closed.

A:

Q: True or False The intersection of two sets is always a subsetof their union

A: To check the statement

Q: Show

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

Q: Prove that if A is transitive set, then A+ is also a transitive set.

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

A:

Q: 2. Specify the set B in set-builder form by giving a written description of its elements, where B =…

A:

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

A:

Q: Give an example of a closed set in R ^ 3 with the usual topology that is not compact

A: Given the usual topology on ℝ3. i.e. τusual=U⊆ℝ3: ∀x∈U, ∃∈>0 such that B∈x⊆U

Q: (d).Show that the derived set E′E′ of any set EE is closed.

A: To show that derived set E′ of any set E is closed.

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

A:

Q: If A is set of all vowels in alphabets.Then number of subsets of A is

A: Given: A contains all vowels  So, A = a, e, i , o , u Number of elements in A = 5

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: 2. Specify the set B in set-builder form by giving a written description of its elements, where…

A:

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: Suppose we modify the construction of the Cantor set by starting with the open interval (0, 1) and…

A:

Q: How to Apply the Archimedes'principle to prove that a set is empty

A: Consider the given information. the Archimedes' principle is defined as, Property states that for…

Q: What is the complement of the set X? How is it denoted?

A: Universal set is always given to find complement of a set. Complement of a set 'X': It includes…

Q: It is the selection of objects from a set.

A: This is the problem of combinatorial. finite math, counting.

Q: Đ A is a closed set.

A: We prove by using the result that intersection of two closed set is closed.

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: 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: 5. Show that a finite set is a closed set.

A:

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

A:

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

A:

Q: let X={a,b,c} Construct two topologies for the set that have at least 4 open and that are…

A: Given that X=a,b,c. To construct two topologies that have at least four open sets. We know that in…

Q: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

A:

Q: {(x,a), (x,b), (x,c), (y,a), (y,b),(y,c)} is the Cartesian product AXB of two sets given as

A: Given,          x,a,x,b,x,c,y,a,y,b,y,c is the Cartesian product of A×B.

Q: Give example for following 1/ set is not compact

A: Compact set:- A set S⊆R is called compact “if every sequence in S has a subsequence that converges…

Q: Suppose that A is the set of sophomores at your school and B is the set of students in discrete…

A: We use set theory here: A: The set of sophomores at your school B: Set of students in discrete…

Q: 2. Prove that by laws of set, (A - B)- (B - C) = A -B

A:

Q: Show that Z has the same cardinality as the set E of even integers.

A: In order to prove that ℤ and set of even integers E have the same cardinality. (i.e) we need to find…

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: 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: prove that every subset of negligible. set is negligible.

A: I have used definition of negligible set.