1. A

    An "optional" proof

    Let A, B⊆ U. Show that A ⊆ A - (B-A). Based on logic alone, I sort of figured out that even if I find (B - A), it would not matter because then I would be left with A - B, correct? However, I could be completely off in that notion.
  2. H

    Linear Algebra, meaning of R^m

    The problem states: "Suppose that {v1,v2,v3} is a linearly independent subset of Rm." What does Rm in this case mean?
  3. A

    Help understanding this proof: E is a subset of f^(-1)(f(E))

    \text{Let } f: X\rightarrow Y, \text{ where } X \text{ and } Y \text{ are sets.} \text{Prove that }E \subseteq f^{-1}(f(E)) \text{ } \forall E \subseteq X. PROOF: \text{Let } x \in E. \text{Then, }y = f(x) \in f(E), \text{ so } x \in f^{-1}(f(E)). \text{ Hence } E \subseteq f^{-1}(f(E)) How...
  4. L

    Need help verifying supremum and infimum of the subset.

    For A = {0,1,2,3,4,5,7,8,9} , let B = {{2,3,4},{0,3,4,5,9},{0,2,3,8}} B is a subset of P(A). Using the partial order ⊆ for P(A) find inf(B) and sup(B). The first thing one must know in order to solve is to know what the spremum and infinmum is. The supremum is the upperbound other wise know as...
  5. S

    Proof of a coset being a subset

    ​Hello all, after wrestling around with this proof for the better part of an hour, I think I got it. If I am wrong, PLEASE do not give the answer! Just a simple hint would be most appreciated. The question is to suppose that H and K are subsets of G, and there are elements a,b in G such that...
  6. S

    Denumerability of a subset of a denumerable subset

    The following is an excerpt from Serge Lang's "Real and Functional Analysis" In the proof, the author defines \left \{k...k_n \right \} as a subset of D. How does he know that D is big enough to contain a set of elements that can be indexed to n? I assume that, by n, the textbook means an...
  7. W

    Prove that each subset of a countable set is countable

    I want to prove: Each subset of a countable set is countable. Here are the definitions available to me: Def: A set A is finite provided A is empty or there is a positive integer N for which there is a 1:1 correspondence between A and the integers 1 through N. A set which is not finite...
  8. B

    Help with a Subset Expression

    Hi, I was hoping somebody could chime in on this problem and confirm my answer. It asks, "Which of the diagrams below, left or right, shows a shaded area that represents the expression: [A U B]^c ∩ C (The ^c is the complement) First image has only C shaded and the second has C, A ∩ C, and B...
  9. A

    Help with proving that kernel is a proper subset of the image of a linear map

    Let $V$ be a vector space of dimension 3, and let $T:V\rightarrow V$ be a nilpotent linear operator of order 3. I need to prove the following: 1. Suppose $v\in V$ is a vector for which $T^2(v)\neq 0$, prove that the set $B=\{v,T(v),T^2(v)\}$ is a basis for $V$. 2. Prove that $Ker(T)\subset...
  10. MechanicalPencil

    Subset of symmetric matricies proof

    Prove that the subset of symmetric (A^T=A ) matrices in Mnxn​ is a subspace. I know the 3 conditions of a subspace, I'm just trying to apply them in a way that satisfies the proof. Am I on the right track? Any help would be appreciated.
  11. A

    Showing each side is a subset of the other side

    I am asked to show that each side is a subset of the other side, but the book's answer is sort of long and hard to understand. Is there another alternative answer that is much shorter and easier to understand? The venn diagram is there just to help me visualize.
  12. F

    Least Common Multiple and All possible subset

    If A={27,42,30,94) is a set,the all possible subsets from the set will be {27},{42},{30},{94},{27,42},{27,30},{27,94},{42,30},{42,94},{30,94}, {27,42,30},{27,30,94},{27,42,94},{42,30,94},{27,42,30,94} .The LCM(least common multiple) for all the asubsets will be 27,42,30,94,378,270,2538,210,1974...
  13. A

    Proper subsets

    Hi all I'm stuck on the question below. I know I have to first prove that C is a subset of D, and then show there is an element of D that doesn't lie in C, but I'm finding it difficult to begin (the little 2 next to R and at the end of the subset for C should be squared). Consider the...
  14. K


    A=(1,2,3.....30) How many 3 element subsets of A are there such that sum of its elements is divisible by 3? Like (1,3,5) or (6,9,3)
  15. A

    A set $A \subset \mathbb{R}^2$ containing more than one point that is invariant under

    A set $A \subset \mathbb{R}^2$ containing more than one point that is invariant under the flow $\phi_t(x,y) = (xe^{-t},ye^{4t})$ A set $A \subset \mathbb{R}^2$ containing more than one point that is not invariant under the flow $\phi_t(x,y) = (xe^{-t},ye^{4t})$ Are these possible? if so...
  16. D

    Subtraction of cardinality of a set and its subset

    Suppose A is finite andB\subseteq A. Prove that |A\backslash B|=|A|-|B|. My attempt: I try to prove it by using induction on B Base case: Suppose B has 1 element, then |A\backslash B|=|A|-1=|A|-|B| I don't think my base case is 'rigorous' enough, it feels there is something missing... For...
  17. L

    List the elements of S produced by the first five applications of the recursive defin

    Let S be the subset of the set of ordered pairs of integers defined recursively by Basis step: (0, 0) ∈ S. Recursive step: If (a, b) ∈ S, then (a + 2, b + 3) ∈ S and (a + 3, b + 2) ∈ S. a) List the elements of S produced by the first five applications of the recursive definition. How do...
  18. A

    Explaining the empty set: subset or element of set

    Greetings, I have a question regarding the difference between an element of a set and a subset, with regards to the empty set. What is the difference between the two statements: (a) Is x an element of {x} and (b) Is x a subset of {x} ? I understand that the empty set is a subset of every set...
  19. M

    Why is the empty subset of a metric space always connected ?

    How can i show that the empty subset of a metric space,X always connected ? It is empty and so will have and so will have an empty boundary.That doesn't seem to be enough. Also, my book says that, no other finite set can be connected,I don't really understand this, because every finite set...
  20. A

    An exercise about limsup/liminf of a subset sequence

    Let E_{n} = $\{$x $\in$ $\[0,2$\pi$]$ : (sin(nx))/n > 0$\}$ with n natural number. Calculate: E', E'' where E' = liminf En, E'' = limsup En for n that goes to $\infty$ Looking at the goniomethrical discus i should say that liminf is the empty set and that limsup is [0,2$\pi$] \\{0,$\pi$...