# nonempty

1. ### Equinumerous + A,B,C are nonempty sets

Let A,B,C are nonempty sets. If A~B, then A x C ~ B x C a) state the converse of this statement b) give an example showing that the converse is NOT always true. THANK YOU!
2. ### S is a nonempty set . A= LUB S. Let T={2x+1:x element in S}.Prove that LUB T=2A+1

Can some one please help me get started on this.I would really like to understand what I am doing. Suppose that S is a nonempty set such that A= LUB S. Let $T={2x+1:x \in S}$. Prove that $LUB T=2A+1$
3. ### To show that the image of a continuous function defined on a compact set is nonempty.

Given a continuous function f defined on a compact set A, to show that f(A) is nonempty, the author of my textbook simply says that "every continuous function defined on a compact set reaches a maximum." Can anyone explain it in more detail? Why is that? Or how should I prove its...
4. ### Sequence within a non-empty, bounded above set

Let S be a non-empty subset of R (real numbers) that is bounded above. Show that there exists a sequence (xn, n is a natural number), contained in S (that is, xn is an element of S for all n in the set of natural numbers) and which is convergent with limit equal to sup S. Any help would be...
5. ### At least two circles tangent to y axis with nonempty intersection

Hi. Here is a problem I've been trying to solve for some time now. Maybe you could help me. We have two sets \mathcal {Q} is a set of those circles in the plane such that for any x \in \mathbb{R} there exists a circle O \in \mathcal {Q} which intersects x axis in (x,0). \mathcal {T} is a set...
6. ### Complement of nonempty open set

I've tried to think this, help would be appreciated to get me going a gain. If we have two nonempty open set A,B \subset {R}^{n} so that A \cap B = \emptyset (two separate sets). How could I show that complement of union A and B is also nonempty, \complement (A \cup B) \neq \emptyset. I'm...
7. ### Revised: A sequence of non-empty, compact, nested sets converges to its intersection

Proposition 2.4.7 * S is a metric space with metric \rho . {A_n} is a sequence of descending, non-empty, compact sets. Then for \epsilon > 0,\ lim A_n = A = \bigcap A_n in the Hausdorff sense. In that sense, one must show that (1) A \subseteq N_\epsilon (A_n)...
8. ### A sequence of non-empty, compact, nested sets converges to its intersection.

A text* I am reading offers a proof that a sequence of non-empty, compact, nested sets converges to its intersection in the Hausdorff metric. I do not follow the second half of the proof which shows that, in the limit, a member of the sequence is contained in the intersection in the sense that...
9. ### Set Theory Problem

A is a convex, nonempty set. A is not bounded below and is bounded above. B* = sup A. Prove: Claim 1: (-inf, b*) is a subset of A Prove: Claim 2: A is a subset of (-inf, b*] Case 1: b* is not an element of A Prove: Claim 2.1: A = (-inf, b*) Case 2: b* is an element of A Prove: Claim 2.2: A =...
10. ### Showing that a non-empty subset M is an affine subset of R^4

Hi, I have to do a project on affine subsets and affine mappings, but I have no clue what they are... We are given only one clue and I can't find many notes on google. I would really appreciate it if someone could help me with this first problem (and if you could also give me a link to some good...
11. ### Non-empty, compact, disconnected and limit points

I am at the moment trying to get through some basic set theory and I'm getting very stuck with the proofs. This question is from a textbook I am studying from and as it is a prove question there is no solution in the back :( Any help with explanations would be very useful Let where E0 = [0...
12. ### Let B={-x : x ϵ A}, A a non-empty subset of R. Show B is bounded below

Hi, I have the following problem... I know the second bit of it, but I have no clue how to prove the first bit. I know B is bounded below (it is basically set A but with opposite signs, so the supremum of A will be the lowest number of set B, its infimum), but I don't know how to express it...
13. ### Prove that G acts transitively on normal subgroup orbits on nonempty finite set A

The Question is let G acts transitively on a nonempty finite set A let H be normal subgroup of G,Let Orbits of H on A be O_1,O_2,...,O_r prove that G acts transitively on O_1,O_2,...,O_r My work I want to prove that for any two oribts O_i,O_j there exist g\in G such that O_i = gO_j I...
14. ### Let ^ be a nonempty indexing set, let A* = {A subscript alpha such that alpha is an..

Let ^ be a nonempty indexing set, let A*={A (subscript alpha) such that alpha is an element of ^} be an indexing family of sets, and let B be a set. Use the results of the theorems for indeing sets and indexing families of sets to prove : B- ( Intersection of (alpha in the ^ <- below intersect...
15. ### Nonempty Subset of Q

Find a nonempty subset of Q (rational numbers) that is bounded above but has no least upper bound in Q. Justify your claim. Thanks for the help!
16. ### An open set is disconnected iff it is the union of two non-empty disjoint open sets

How do I proof this is the definition of disconnectedness that I am using that S is disconnected if there exists two sets S_{1}, S_{2} such that they are both non-emtpy, S_{1} \cup S_{2} = S and cl(S_{1})\cap S_{2} = cl(S_{2})\cap S_{1} = \emptyset Attempting to prove the first condition, that...
17. ### Any set of m positive integers contains a nonempty subset whose sum is ...

Any set of m positive integers contains a nonempty subset whose sum is a multiple of m. Proof. Suppose a set T has no nonempty subset with sum divisible by m. Look at the possible sums mod m of nonempty subsets of T. Adding a new element a to T will give at least one new sum mod m, namely the...
18. ### nonempty family sets

Let A be any NONEMPTY family sets. Prove \bigcap_{A \epsilon A} A \subset \bigcup_{A \epsilon A} A .
19. ### Closed, non-empty and bounded sets.

Suppose that for each n \in \mathbb{N} we have a non-empty closed and bounded set A_n \subset \mathbb{C} and A_1 \supseteq A_2 \supseteq ... \supseteq A_n \supseteq A_{n+1} \supseteq ... Prove that \bigcap_{n=1}^{\infty} A_n is non empty. [Hint: use Bolzano-Weierstrass] Solution...
20. ### Proof of non-empty subsets, glb and lub

Let S be a non-empty subset of R and Suppose there is a non-empty set of T subset of S. a) Prove: if S is bounded above, then T is bounded above and lub t< or = lub S. b) Prove: if S is bounded below, then T is bounded below and glbT is > or = glbS. I am having a really hard time with this.