1. H

    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. A

    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. R

    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. S

    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. W

    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. M

    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. G

    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. G

    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. K

    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. J

    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. K

    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. J

    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. Amer

    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. B

    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. J

    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. Pinkk

    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. O

    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. O

    nonempty family sets

    Let A be any NONEMPTY family sets. Prove \bigcap_{A \epsilon A} A \subset \bigcup_{A \epsilon A} A .
  19. slevvio

    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. J

    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.