Sets Definition and 1000 Threads

Homework Statement Let f be a function from E to F . Prove that f is an injective function if and only if for all A and B subsets of P(E)^2. f(A\cap B)=f(A)\cap f(B) The Attempt at a Solution Well since we have "if and only if" that means we have an equivalences so for. \Rightarrow If f...

I know this is probably the most basic question imaginable so please bear with me. I did google it but I still couldn't figure it out. Say you have a function f(x, y, z) and a point (x_0, y_0, z_0) that satisfies the equation f(x, y, z) = 0. Does that imply that f(x_0, y_0, z_0) \in f(x, y, z)...

Definition: A subset $D$ of $\mathbb R$ is said to be discrete if for every $x\in D$ there exists $\epsilon>0$ such that $(x-\epsilon,x+\epsilon)\cap D=\{x\}$. Question: Does there exist a discrete subset $D$ of $\mathbb R$ such that the set of limit points of $D$ is an uncountable set. ___...

I would like to know if there is a general formula, and if so, what it is, for finding the $limsup$ and $liminf$ of a sequence of sets $A_n$ as $n\rightarrow \infty$. I know the following examples: **(1)** for $A_n=(0,a_n], (a_1,a_2)=(10,200)$, $a_n=1+1/n$ for $n$ odd and $a_n=5-1/n$ for $n$...

Prove that for any collection {Oα} of open subsets of ℝ, \bigcap Oα is open. I did the following for the union, but I don't see where to go with the intersection of a set. Here's what I have so far: Suppose Oα is an open set for each x \ni A. Let O= \bigcap Oα. Consider an arbitrary...

Homework Statement I need to answer a bunch of topological questions based on the cartesian product of two sets, but I'm not entirely sure how to graph them out. I have A = [1,2)U{3} and B = {1, (1/2), (1/3), ...}U[-2,-1). S = A x B, and I need the graph of S. Could anyone help me with...

In a book I'm reading it says: \newline If f: \mathbb{R} \longrightarrow \mathbb{R} is lower semi continous, then \{f > a \} is an open set therefore a borel set. Then all lower semi continuous functions are borel functions. It's stated as an obvious thing but I couldn't prove it. The definition...

How many elements does each of these sets have where a and b are distinct elements? (with steps please) a) P({a,b{a,b}})b)P({∅,a,{a},{{a}}}) c)P(P(∅)) *i have tried to solve them but i am a little bit confused... Thanks in advance :)

Obviously countable sets of R are Lebesgue null sets - but are Lebesgue null sets countable?

Suppose that H, K are Hilbert spaces, and A : H -> K is a bounded linear operator and an isomorphism. If X is a dense set in H, then is A(X) a dense set in K? Any references to texts would also be helpful.

Homework Statement Theorem: Let S be a subset of the metric space E. Then S is closed iff whenever p1,p2,p3,... is a sequence of points of S that is convergent in E, we have lim n→∞ p_n ∈ S. Homework Equations The Attempt at a Solution I am having trouble understand the "if"...

Homework Statement . Let ##f: (X,d) → (Y,d')## a uniform continuous function, and let ##A, B \subseteq X## non-empty sets such that ##d(A,B)=0##. Prove that ##d'(f(A),f(B))=0## I've been thinking this exercise but I don't have any idea where to or how to start, could someone give me a hint?

Homework Statement I am very uncomfortable with the concept of "level sets" and don't quite get what it means. Homework Equations y = 2(x1)^2 - (x1)(x2) + 2(x2)^2 y = 2x1^(1/2) * x2^(1/2) The Attempt at a Solution I'm not even sure where to start... Can anyone please give me...

Hi everyone, new member here. Anyway, for my astronomy class my professor wants us to calculate the zone time the center of the sun sets on the winter solstice, at 40 degrees north latitude, and I'm a bit stuck. I know on the winter solstice, the declination of the sun is -23.5 degrees and the...

I have a (probably trivial) question about coordinate charts. I've been studying Sean Carroll's lecture notes on General Relativity. I'm on my second re-read and I'm trying to make sure I understand the basics properly. I hope the terminology is correct - this is my first use. Carroll cites...

1. Homework Statement . Let A be a chain and B a partially ordered set. Now let f be an injective function from A to B and suppose that if a,b are elements of A and a≤b, then f(a)≤f(b). Prove that f(a)≤f(b) implies a≤b. 3. The Attempt at a Solution . I want to check if this proof by...

open

prove the followinga. prove that if $A\cap B=\emptyset$, then $(A\times C)\cap (B\times C)=\emptyset$ b. $A\cup(B\cap C)=(A\cup B)\cap (A\cup C)$ i don't have any idea how i would start proving this. can you give me some techniques on proofs.

I've been struggling for a few minutes with this basic thing and I want to make sure I got it right, given A, B being disjoint, We know that P(A and B) = 0 However, if they are independent then P(A and B) = P(A) x P(B) Then if P(A) is finite non zero and P(B) is finite non zero, how...

the associative axioms for the real numbers correspond to the following statements about sets: for any sets A, B, and C, we have $(A\cup B)\cup C=A\cup (B\cup C)$ and $(A\cap B)\cap C=A\cap (B\cap C)$. Illustrate each of these statements using Venn diagrams. can you show me how to draw the...

My basic question is this: does an arbitrary open set in ℝ2 look like a bunch of regions bounded by continuous curves, or are there open sets with weirder boundaries than that? Let me state my question more formally. A Jordan curve is a continuous closed curve in ℝ2 without self-intersections...

Written by micromass: The newest challenge was the following: This was solved by HS-Scientist. Here's his solution: This is a very beautiful construction. Here's yet another way of showing it. Definition: Let ##X## be a countable set. Let ##A,B\subseteq X##, we say that ##A## and...

The axiom of choice on a finite family of sets. I just been doing some casual reading on the Axiom of CHoice and my understanding of the is that it assert the existence of a choice function when one is not constructable. So if we have a finite family of nonempty sets is it fair to say we can...

i have solved these problem just want to make sure I'm on the right track. 1. Say the football team F, the basketball team B, and the track team T, decide to form a varsity club V. how many members will V have if $n\left(F\right)\,=\,25,\,n\left(B\right)\,=\,12,\,n\left(T\right)\,=\,30$ and no...

just want to know if my answers are correct. 1. for any set A, a set of subsets of A is said to be exhaustive if the union of these subsets is A, and is said to be disjoint if no two of the subsets have any element in common. if $\displaystyle A\,=\,\{a,\,b,\,\,c\},\,$ tell whether the...

just want to make sure if my answer is correct If $\displaystyle U\,=\,\{0,\,1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,9\}$, the set of digits in our decimal system, and $\displaystyle A\,=\,\{0,\,1,\,2,\,3,\,4,\,5\}$, $\displaystyle B\,=\,\{2,\,3,\,4,\,5,\}$, $\displaystyle C\,=\,\{4,\,5,\,6,\,7\}$...

please help me understand what my book says: If set A has only one element a, then $\displaystyle A\,x\,B\,=\, \{\left(a,\, b\right)\,|\,b\,\epsilon\,B\}$, then there is exactly one such element for each element from B. can you explain what it means and give some examples. thanks! :)

Homework Statement Let A be a set, prove that the following statements are equivalent: 1) A is infinite 2) For every x in A, there exists a bijective function f from A to A\{x}. 3) For every {x1,...,xn} in A, there exists a bijective function from A to A\{x1,...xn} Relevant...

I have two questions explain why any subset of a finite set is finite. (prove) and why is empty set is considered to be a subset of any set? I'm confused, because let's say set A is a subset of set B it means that every element of A is an element of B. in the case of empty set being a...

R1 = { (1,2) , (1,3) , (1,4) (1,5) , (1,6) } No R2 = { (x,y) in R x R | x = sin(y) } R3 = { (x,y) in Z x Z | y2 = x } R4 = { (Φ, {Φ}) , ({Φ},Φ) , (Φ,Φ) , ({Φ},{Φ}) } No R5 = { (x,y) in N x Z | 0<x<1, 3<y<4 } A x B means Cartesian product. That much I know. What I don't know is how to...

I'm wondering if the following is true: Every closed subset of ##\mathbb{R}^2## is the boundary of some set of ##\mathbb{R}^2##. It seems false to me, does anybody know a good counterexample?

I was just wondering, is it possible? It's regarding a debate on whether mathematics is an invention or discovery.

In the Principles of Mathematical analysis by Rudin we have the following theorem If \mathbb{K}_{\alpha} is a collection of compact subsets of a metric space X such that the intersection of every finite sub collection of \mathbb{K}_{\alpha} is nonempty , then \cap\, \mathbb{K}_{\alpha} is...

In reading about the Tube Lemma, an example is given where the Tube Lemma fails to apply: namely, the euclidean plane constructed as R X R. The Tube Lemma does not apply here because R is not compact. The example given is as follows: Consider R × R in the product topology, that is the...

Here is the question: I have posted a link there to this topic so the OP can see my work.

My roku device has a channel for wunderground.com, which shows the weather. But it also shows a time-lapse video of the past 24 hours in a local neighborhood. The caption says "Facing West," and you can indeed see the sun setting in the evening as you would expect. But in the video the sun...

I have a set of data that was recorded from an engine that we are testing. We've noticed lately that a particular pressure value will sometimes spike with no apparent explanation, as seen in the attached graph. The pressure in question is passively regulated by a pump, but it is also dependent...

I have begun to learn about maximal elements from a linear algebraic perspective (maximal linearly independent subsets of vector spaces). I have a few questions of which I have been able to get few insights online: 1) Does every family of sets have a maximal element? How can I make a family...

Problem: Prove that if an element is in the union of two infinite sets then it is not necessarily in their intersection: Proof: Have I solved it correctly?

Class of all finite sets In a higher algebra book that I'm working through, the natural numbers are constructed in the following manner:- Consider the class S of all finite sets. Now, S is partitioned into equivalence classes based on the equivalence relation that two finite sets are...

If you unite infinitely many open sets you still get an open set whilst the same is not necessarily true for a closed set. Can someone try to explain what property of a union of open sets it is, that assures that an infinite union is still open (and what property is the closed sets missing?)

The set of all functions is larger than 2^{\aleph_0} . So let's say I wanted to average over all functions over some given region. that was larger than 2^{\aleph_0} how would I do that.

1. Homework Statement Let A and B be nonempty bounded subsets of \mathbb{R}, and let A + B be the set of all sums a + b where a ∈ A and b ∈ B. (a) Prove sup(A+B) = supA+supB .Homework Equations The Attempt at a Solution Let Set A=(a_1,...,a_t: a_1<...a_i<a_t) and let set B=(b_1,...,b_s...

Homework Statement Prove that a finite union of countable sets is also countable. Is an infinite union of countable sets also countable?Homework Equations A set S is countable if and only if there exists an injection from S to N.The Attempt at a Solution I will attempt prove it for the case of...

Is it possible to define sets from just the peano axioms? Usually when people use the peano axioms as the basis of their math they just assume the existence of sets but without axioms regarding sets we technically shouldn't just say they exist. Oh, also there are two versions of the...

Hello, I am having some difficulties understanding why a subset under the usual metric topology of the reals is connected. How can a set X = (0,1] u (1,2) be connected? The definition I am using is: A is disconnected if there exists two open sets G and V and the following three properties...

Let $X$ and $Y$ be any two sets. Show by using Zorn's Lemma (or anything equivalent to it) that there is an injection of one into the other.

Homework Statement Find the error in this proof and give an example in (ℝ,de) to illustrate where this proof breaks down. Proof that every totally bounded set in a metric space is bounded. The set S is totally bounded and can therefore be covered by finitely many balls of radius 1, say N...

Suppose you have two sets S_{1} and S_{2}. Suppose you also know that every vector in S_{1} is expressible as a linear combination of the vectors in S_{2}. Then can you conclude that the two sets span the same space? If not, what if you further knew that every vector in S_{2} is expressible...

I have 17 elements, and I want to put them in 3 sets. This makes 2 sets with 6 elements, and 1 set with 5 elements. Now I want to find an algorithm to partition n elements in k sets. Can anyone give me an algorithm, or a math expression for me to implement in a algorithm? Thanks