Sets Definition and 1000 Threads

Hello! (Wave) Could you give me a hint how I can show that the sets: $$\{ \varnothing, \{ \varnothing \} \} , \ \ \ \{ \varnothing, \{ \varnothing, \{ \varnothing \} \} \}, \ \ \ \{ \{ \varnothing \}, \{ \varnothing, \{ \varnothing \} \} \}$$ are pairwise disjoint=are not equal ? :confused:

Let a, b, c, and d be real numbers with a < b < c < d. Express the set [a, b]U[c, d] as the difference of two sets. I know that [a,b]U[c,d] is a union and what a difference of two sets is, but I don't quite understand this question.

Could someone please explain how the image of a set A' ⊆ A is the set: f(A') = {b | b = f(a) for some a ∈ A'}. And how can the complement of A be a subset of A? Forgive my ignorance here, I'm a beginning student of set theory. Edit: Maybe I should rephrase my question: Could you explain what...

Hi , Can anyone please give me an idea to disprove the following with counter example: A , B & C be sets. If A X C = B X C , the A = B . I tried giving random numbers in venn diagram but didn't work. And, using subset way to prove equal but still couldn't solve it.

How are you supposed to go about putting together the power set of a set of sets such as X = {{1},{1,2}} What is the power set of X then? And what's the rule for calculating cardinality for the power set of a set that consists of elements which are sets such as the above? Because the set X...

What exactly prevents us from ruling out a uniform distribution on infinite sets? To be more precise, why are distributions and limits like \int_{-\infty}^{+\infty}dx\,\lim_{\sigma\to\infty}f_{\mu,\sigma}(x) = 1 \int_{-\infty}^{+\infty}dx\,\lim_{\Lambda\to\infty}\frac{1}{\Lambda} \chi_{[a,a+L]}...

Homework Statement Suppose R is a relation on A, and define a relation S on P (A) as follows: S = {(X, Y ) ∈ P (A) × P (A) | ∀x ∈ X∃y ∈ Y (xRy)}. For each part, give either a proof or a counterexample to justify your answer. (a) If R is reflexive, must S be reflexive? (b) If R is symmetric...

Ok. I don't understand sets within sets. No one has been able to explain it to me simply. So you can have sets inside of sets. That I get. But what happens when you get to something tricky like the empty set? For example, I know that ø ε {ø} is true. I don't understand how. How do I read...

Homework Statement I am struggling with what seems like a very simple problem from Terrence Tao's Introduction to Measure Theory book (which is available for free online by the way). What I am trying to prove is the following: Give an alternate proof of Lemma 1.1.2(ii) by showing that any two...

Homework Statement Suppose A is a set, and for every family of sets F, if ∪F = A then A ∈ F. Prove that A has exactly one element. (Hint: For both the existence and uniqueness parts of the proof, try proof by contradiction.) Homework Equations The Attempt at a Solution Let A be...

To prove that two sets are in fact the same, do I actually have to prove that the two are subsets of each other; or could I prove that they are equivalent by some other means, such as invoking the definitions of the sets? For instance, the I am trying to show that the binary set operator...

Hello all, I was wondering how to make that nice looking Complex 'C' when writing a set in Latex. For example, $$\{k\in C\mid k>0\}$$ looks okay but can be better.

Homework Statement . Let ##X## be a nonempty set and let ##x_0 \in X##. (a) ##\{U \in \mathcal P(X) : x_0 \in U\} \cup \{\emptyset\}## is a topology on ##X##. (b) ##\{U \in \mathcal P(X) : x_0 \not \in U\} \cup \{X\}## is a topology on ##X##. Describe the interior, the closure and the...

I am reading Chapter 2: Vector Spaces over $$\mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C}$$ of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding infinite direct sums and products and indexed families of sets ... ... On page 62, Knapp introduces direct...

On a post involving the proof of the Fourth Isomorphism Theorem for vector spaces (in which I was immeasurably helped by Deveno) I have become aware that my knowledge of sets and functions was not all it should be when it comes to things like inverse images, left and right inverses and the like...

Consider the sets ##X:= \{x\in\mathbb R^2: \enspace ||x-(-1,0)||_2 \leq 1\}## (a ball) and ##Y:=co\{(0,-1), (0,1), (1,0)\}## (a triangle). Both ##X## and ##Y## are compact and convex, but they aren't disjoint: ##X\cap Y = \{(0,0)\}##. Since they aren't disjoint, the most common separating...

Hello, my problem is the following: A lasers gives out a bunch of data points which are reflected off a metal surface and recorded by a camera attached to the side of the laser. The image the camera receives is however distorted. In order to calibrate the camera I need to find a function...

Homework Statement A study was done to determine the efficacy of three different drugs – A, B, and C – in relieving headache pain. Over the period covered by the study, 50 subjects were given the chance to use all three drugs. The following results were obtained: 21 reported relief from drug...

Let X be an arbitrary set and P(X) the set of all its subsets, prove that if ∀ A,B ∈ P(X) the sets A∩B,A∪B are also ∈ P(X). I really don't know how to get started on this proof but I tried to start with something like this: ∀ m,n ∈ A,B ⇒ m,n ∈ X ⇒ Is this the right way to start on this proof...

Homework Statement Let A be the 4x2 matrix |1/2 -1/2| |1/2 -1/2| |1/2 1/2| |1/2 1/2| Find the projection matrix P that projects vectors in R4 onto R(A) Homework Equations projSx = (x * u)u where S is a vector subspace and x is a vectorThe Attempt at a Solution v1 = (1/2, 1/2...

Homework Statement Prove A \times (B \cap C) = (A \times B) \cap (A \times C) The Attempt at a Solution Let x \in A and y \in B \cap C \rightarrow y \in B \wedge y \in C now \exists (x,y) \in A \times (B \cap C) so (x,y) \in A \times B \wedge (x,y) \in A \times C thus...

Hi, I want to show that there exists a well ordering for every finite set. (I know if you add axiom of choice you can prove this theorem for infinite sets too but I think the finite sets do not need axiom of choice to become well ordered)

I think this looks like a homework problem, so I'll just put it here. Homework Statement Demonstrate that, for any index category ##\mathscr{J}## and any diagram ##\mathcal{F}:\mathscr{J}\to\mathbf{Sets}##, $$\varprojlim_{\mathscr{J}}A_j=\left\{a\in \prod_{j\in \operatorname{obj}(...

Homework Statement Let ##X## be a topological space. Let ##A_1 \supseteq A_2 \supseteq A_3...## be a sequence of closed subsets of ##X##. Suppose that ##a_i \in Ai## for all ##i## and that ##a_i \rightarrow b##. Prove that ##b \in \cap A_i##. Homework Equations The Attempt at a Solution...

Homework Statement Show that for a set A\subsetN, which is numerically equivalent to N=Z+, and the set B = A \cup{0}, it holds that A and B are numerically equivalent, i.e., that A \approxB Hint: Recall the definition of A≈B and use the fact that A is numerically equivalent to N. Note...

Homework Statement I'm trying to understand what compact sets are but I am having some trouble because I am having trouble understanding what open covers are. If someone could reword the following definitions to make them more understandable that would be great. Homework Equations...

Set Theory -- Uncountable Sets Homework Statement Prove or disprove. There is no set A such that ##2^A## is denumberable. The Attempt at a Solution A set is denumerable if ##|A| = |N|## My book shows that the statement is true. If A is denumerable, then since ##|2^A| > |A|, 2^A ##...

Homework Statement Explain why ## (0,1 ) ## and ## (0,2)## I have proved that a function ## f: (0,1 ) -> (0,2) ##defined by f(x) =2x is bijective.Homework Equations The Attempt at a Solution I could state that due to the sets relationship being bijective they have the same cardinality, but...

My question is just to ask whether the operations like:- AUB is a relation or not? in our book it is written that the relations of two sets should be subset of the cartesian product of two sets but i think that relations are those which connects two sets and that can be AUB(A union B)...

I was trying to prove: d(A,B) = d( \overline{A}, \overline{B} ) I "proved" it using the following lemmas: Lemma 1: d(A,B) = \inf \{ d(x,B) \}_{x \in A} = \inf \{ d(A,y) \}_{y \in B} (By definition we have: d(A,B) = \inf \{ d(x,y) \}_{x \in A, y \in B} ) Lemma 2: d(x_{0},A) = d(x_{0}...

show that in a set of any 5 consecutive numbers there is at least one number that is co-prime to all the rest 4 (for example (2,3,4,5,6- 5 is co-prime to 2,3,4,6)

Hi I have two sets of data, y as a function of x, for two different experiments. Y is the dependant variable, an instrument reading, x is the independant variable, a concentration of something in a solution. Plotting the data in excel gives me 2 curves. The experiments were...

Homework Statement I'm trying to do a problem, and in order to do it I need to find a function f:R→R which is continuous on all of R, where A\subseteqR is open but f(A) is not. Can anyone give an example of a function that satisfies these properties? I think once I have an example I'll...

A few years back I started a thread to make the point that there is a common misconception about main-sequence stars that their fusion rate sets their luminosity, in the sense that to know what the luminosity of the star will be, you need to know what the fusion rate is. In particular, you...

If I have finite sets X,Y, and need to prove that X ⊆ Y <=> P(X) ⊆ P(Y), where P() denotes the power set of a set. I started out saying that for infinite sets X,Y, x⊆X, and y⊆Y. Given that X⊆Y, we want to show that P(B)⊆P(Y). x⊆X, so through transitivity, x⊆Y (is this correct?). From here, I...

Hi all. I'm having trouble understanding the cartesian product of a (possible infinite) family of sets. Lets say \mathcal{F} = \{A_i\}_{i \in I} is a family of sets. According to wikipedia, the cartesian product of this family is the set \prod_{i \in I} A_i = \{ f : I \to \bigcup_{i...

Homework Statement Prove that the union of a collection of indexed sets has finite diameter if the intersection of the collection is non-empty, and every set in the collection is bounded by a constant A. The Attempt at a Solution The picture I have is if they all intersect (and assuming...

We have this set of primes which is infinite. This has lots of different subsets. Here is the list of subsets: Real Eisenstein primes: 3x + 2 Pythagorean primes: 4x + 1 Real Gaussian primes: 4x + 3 Landau primes: x^2 + 1 Central polygonal primes: x^2 - x + 1 Centered triangular primes: 1/2(3x^2...

Hi everyone, a couple of technical questions : 1) Definition: Anyone know the definition of the induced orientation of a submanifold S of an orientable manifold M? 2)Dividing sets in contact manifolds: We have a contact 3-manifold (M3,ζ ). We define a surface S embedded in M3 to be a convex...

Let $$(E=]-1,0]\cup\left\{1\right\},d) $$ metric space with $$d$$ metric given by $$d(x,y)=|x-y|$$, and $$||$$absolute value. How I can find open sets of E explicitly? Thanks in advance.

Cartesian product of indexed family of sets The definition of a Cartesian product of an indexed family of sets (X_i)_{i\in I} is \Pi_{i\in I}X_i=\left\{f:I \rightarrow \bigcup_{i \in I} \right\} So if I understand correctly, it's a function that maps every index i to an element f(i) such...

The problem statement Let ##\mu## be a measure defined on the Borel sets of ##\mathbb R^n## such that ##\mu## is finite on the compact sets. Let ##\mathcal H## be the class of Borel sets ##E## such that: a)##\mu(E)=inf\{\mu(G), E \subset G\}##, where ##G## is open...

We define by recursion the set of sets {An:n∈ℕ} this way: A_0=∅ A_n+1=A_n ∪ {A_n}. I want to prove by induction that for all n∈ℕ, the set A_n has n elements and that A_n is transitive (i.e. if x∈y∈A_n, then x∈A_n). My thoughts: for n=0, A_1 = ∅∪ {∅} = {∅} then, for n+1: A_n+2...

Homework Statement I have 2 sets, one with 5 elements (A) and the other two(B). How many onto functions can be made from A to B? Homework Equations The Attempt at a Solution My first thought is that it should be something like ##\frac{5!}{2!}=60##. I don't know if this is correct...

Homework Statement Determine if the sets A, B, C, and D satisfy the following proposition (p) for the set S: Homework Equations p: for all ε > 0, ∃ x \in S such that x < ε A = {1/n : n \in Z+} B = {n : n ε Z+} C = A \cup B D = {-1} The Attempt at a Solution I am...

Show that the sets $$\{a,b\}$$ and $$\{a, b, a-b\}$$ of real vectors generate the same vector space. How to proceed with it? I guess the following expression is helpful. $$c1*a+c2*b+c3*(a-b)=(c1+c3)*a+(c2-c3)*b=k1*a+k2*b$$

If $G$ and $\left\{F(n): n \in \mathbb{K}\right\}$ are a family of sets, show that $\displaystyle G \cap \cap_{n \in \mathbb{K}}F(n) = \cap_{n \in \mathbb{K}}(F(n) \cap B).$ I said if $b$ is an ement of $\displaystyle G \cap \cap_{n \in \mathbb{K}}F(n)$ then $b$ is in both $G$ and $F(n)$ for...

I'm having trouble with the following: Let R be a relation on A. Prove that if Dom(R) \bigcap Range(R) = ø, then R is transitive. I took the negation of the "R is transitive" to try proof by contrapositive (as the professor suggested), and have the following: \exists x,y,z \in A s.t. (x,y)...

Homework Statement For every pair of sets (A,B) we have P(AxB)=P(A)xP(B) Prove or disprove the above statement. Homework Equations The Attempt at a Solution I have attempted solving this using A={1,2} and B={a,b} AxB={(1,a),(1,b),(2,a),(2,b)}...

Homework Statement Homework Equations The Attempt at a Solution $$(A-B)\cup (C-B)=(A\cup C)-B\\ (A\cap B^{ C })\cup (C\cap B^{ C })=(A\cup C)\cap B^{ C }\\ (A\cup C)\cap B^{ C }=(A\cup C)\cap B^{ C }\\$$ I know for algebraic proofs, proofs like these are accepted if they are reversed. But...