Question

Asked Oct 27, 2019

59 views

Step 1

(a)

To prove: - Intersection of convex set is convex.

Let us take two convex set (let A and B) and C=A∩B

From the definition convex set, a subset S of a plane satisfying incidence and betweenness axioms is called convex if for any two points a, b in S the whole segment ab is contained in S.

Let x and y be two points in C then the segment xy belongs to both the set that is, in set A and set B. It is already known that both the set A and B are convex.

Hence segment formed by every pair of point in the set C is contained in the set C. Hence, we conclude that the set is convex.

Similarly, for the family of convex sets, the line segment formed by any two points in the intersection of the sets belongs to every convex set of the family which implies that the intersection of (any family of) convex sets is convex.

An example of the intersection of two circular disks is shown in the figure: -

In this figure, segment formed by any two points in C is contained in C.

Step 2

(b)

To prove: Any ray is convex set.

Let us consider a ray say AB and let us take any two arbitrary points C and D on the ray AB.

The line segment CD completely lies on the ray AB and from the definition of convex set, a subset S of a plane satisfying incidence and betweenness axioms ...

Tagged in

Find answers to questions asked by student like you

Show more Q&A

Q: Use this in Exercises 16-21 to find a particular solution . Then find the general solution and, wher...

A: The given differential equation is

Q: Exercise 6:(a) Suppose lim_>c f(x) = L. Show that for every M L, there exists & > 0 such that f(x) M...

A: a.Proof:

Q: How many quarts can be measured from 2.5 kilograms of honey?

A: Recall the following fact.

Q: 2. An inner product on a vector space V is any mapping ,-) V x V R with the following properties: i....

A: We know that

Q: Is a permutation of one element considered to be an even permutation? For example, is (1) an even pe...

A: To clarify the notion of an even permutation

Q: Suppose that you have a covering of a sphere by finitely many( closed) hemispheres. Prove that there...

A: Given that the sphere is covering by finitely many (closed) hemispheres.

Q: In Exercises 18-30 find a fundamental set of solutions, given that yı is a solution. х2у" — 2ху' + (...

A: Click to see the answer

Q: How do I solve the simultaneous following equations using matrix algebra?x + y + z = 3 x + 2y - z = ...

A: To apply matrix method to solve the given simultaneous equations

Q: How do you solve this?

A: Add the slack variable S1 in the constraint -1 and add slack variable S2 in the constraint -2. After...