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

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: Suppose that |A| = |B| = |C| = 6 and |A ∩ B| = |A ∩ C| = |B ∩ C| = 2. Determine the smallest that |A…

A: Given, |A| = |B| = |C| = 6 and |A ∩ B| = |A ∩ C| = |B ∩ C| = 2

Q: True or false? For all sets A and B,  (A∪B)'=A'∪B' This can be determined by constructing a…

A:

Q: (b) Show that the set A = {(x, y) E R² : y > 0} is not complete.

A: Given: The given set ,

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

A: To check the statement

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

A:

Q: O Show that if Si and S2 are convex sets in R™X", then so is t mxn S = {(x, y1 + y2) | x € R", y1,…

A: Convex set: , If a and b are points in a vector space S. Then the vector space is called convex set…

Q: What is the intersection of sets A and B?

A: The intersection of sets is a set which include the common of the two sets A and B. It is…

Q: 11 - Which of the following statement regarding sets is false I. AnA=A AUA = A A- (Bn C) = (A – B) U…

A: Union and intersection apply on same set gives the set itself

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: Đ A is a closed set.

A:

Q: Is it true that for any sets A and B,        P ( A ) ∪ P ( B ) = P ( A ∪ B )? Justify your answer.

A:

Q: - If C is Convex set ond a GR, then XC is a Convex set

A: Let C be a convex set and x∈R Then we have to show that xC is a convex set. The set xC is defined as…

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: Suppose that you have two sets, and you know that both sets are convex. Proven that the intersection…

A: convex set is a set of points such that, given any two points x,y in that set, the straight line…

Q: _1. It is a pictorial representation of the relationships between sets wherein the sets are…

A: Since you have posted multiple questions. We will solve first question. If you want the other…

Q: Are the sets A_4 and A_5 compact sets? Why yes, why not? Ay = (,2] A- [1, 2] u [ 3, +]

A: See the attachment

Q: Consider whether the following statements are true or false: a. The intersection of any two convex…

A: To state true or false

Q: (10) Show that the set A = {a € Q : a² < 2 or a < 0} is a %3D cut.

A:

Q: Prove that a closed convex set in R" has a finite number of extreme points if and only it is the…

A: A closed convex set in ℝn has a finite number of extreme points if and only if it is the…

Q: 17. Which of the following accurately defines the complement of a given set, say A? A. {x|x€U and…

A: Complement of any given set is defined by all the elements in the universal set U which do not…

Q: If G(sub alpha) is a collection of closed sets, then the intersections of the G(sub alphas) is a…

A:

Q: Show that the following pair of sets have the same cardinality. A = (5, 9] and B = (1, 2) ∪[4, 7]

A: Two sets A and B have the same cardinality provided there is a bijection from A to B.

Q: a. Show that the null-space N(A) is a convex set. Is it true for every A?

A: As both questions are of different type,we shall solve first Que only. For other questions kindly…

Q: Let C be a nonempty subset of a bounded set D. Why does inf C and sup C exist? Show that (c) inf D ≤…

A:

Q: A set L cR is convex if:

A: In this question, we have to find the correct option for the convex set L⊂Rn  here we use the direct…

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: Let A, B and C be non-empty sets and let X= (A-B)-C and Y=(A-C)-(B-C) .Check whether X=Y or not.

A:

Q: If A is a countable set and B is a subset of A then

A: If A is a countable set and B is a subset of A then B is also a countable set since, every subset of…

Q: b) B = {(x, y) : x² < e*+3y < 5y + 4x},

A: The objective is to find the closure of the given set.

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 the following:Suppose thatX is a tidy set. Then for any sets a and b, if a ∈ b and b ∈ X, then…

A: Consider,X be a tidy set that represent all data set that each column is a variable and each raw is…

Q: 7. Show that the Obenchain class LOb(B) is a convex set.

A: Given: To show the Obenchain class LOb (β) is a convex set is given as,

Q: 5. If X any Y are sets, then X-Y {r€X: z4Y} If X is open set and Y is closed set. Prove that (a) X-Y…

A: Let X is open set and Y is closed set

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: is the set [5,100] compact?

A: Given that:A set is S=5,100.We have to check whether a set 5,100 is compact or not?

Q: If X and Y are two sets such that X ∪ Y has 50 elements, X has 28 elements and Y has 32 elements,…

A: Given that n(X ∪ Y)= 50  n(X)= 28 and n(Y)=32  We have to find n(X ∩ Y )= ?

Q: What will be the cartesian product of three sets if one set of them is empty?

A: The answer is given as follows:

Q: If S1 and S2 are convex sets, prove that their intersection S1 N S2 is also a convex set.

A:

Q: Show the following set is convex. x≤10 [(1+(x-10)-²)¹ x>10 Determine a-cut sets of the above set for…

A: To show that the given set is convex: μAx=0                               x≤101+x-10-2-1  x&gt;10…

Q: Show that the feasible set X = {r: g1(x) < 0,..., gr(x) <0} is a convex set, if all the functions 9;…

A:

Q: 2. If A, B and C are sets, use a membership table to prove or disprove: A U (B N c) = (A U B) n (A U…

A: First we prove that left hand side is subset of right hand side and then right hand side is subset…

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: 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: 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:

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

A: I have used definition of negligible set.

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”.