Sets Definition and 1000 Threads

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\...

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

Here is the question: Here is a link to the question: Abstract math question: bijectivity on finite and infinite sets? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

my knowledge of time-space is limited, so my question might be poorly/wrongly constructed/verbalized: Do we have two sets of co-ordinate systems when space-time is bent (by say, mass)? in one system the circle becomes, say, an ellipsoid while in other it remains a circle? in one...

What is a countable set exactly? HELP? Can someone help guide me through this problem? I'm a bit lost on how to show this... Countable union of countable sets: Let I be a countable set. Let Ai , i ∈ I be a family of sets such that each Ai is countable. We will show that U i ∈ I Ai is countable...

Homework Statement Let S = {(a,b) : 0 < a < b < 1 } Union {R} be a base for a topology. Find subsets M_1 and M_2 which are compact in this topology but whose intersection is not compact. Homework Equations The Attempt at a Solution I'm not even sure what it means for an element of S to be...

Homework Statement Hi, This is my first post. I had a question regarding open/closed sets and subspace topology. Let A be a subset of a topological space X and give A the subspace topology. Prove that if a set C is closed then C= A intersect K for some closed subset K of X. Homework...

Homework Statement The question is: Let ##\pi=\left \{ x\in\mathbb{R}^n\;|\;x=(x_1,...,x_{n-1}, 0) \right \}##. Prove that if ##E\subset\pi## is a closed Jordan domain, and ##f:E\rightarrow\mathbb{R}## is Riemann integrable, then ##\int_{E}f(x)dV=0##. Homework Equations n/a...

The question is:Let $\pi=\left \{ x\in\mathbb{R}^n\;|\;x=(x_1,...,x_{n-1}, 0) \right \}$. Prove that if $E\subset\pi$ is a closed Jordan domain, and $f:E\rightarrow\mathbb{R}$ is Riemann integrable, then $\int_{E}f(x)dV=0$.(How to relate the condition it's Riemann integrable to the value is $0$...

Let f be a continuous function from R to R and let A be a subset of R^2. Define A={(x,y): y<f(x)}. Can you express A as a cartesian product of two open sets? I tried RxU alpha_x where alpha_x = {y:y<f(x)}. But that didn't work, i need to change something about R.

Theorem: Let S be a compact subset of ℝ^n. Then S is closed. Before looking at the book I wanted to come up with my own solution so here is what I've thought so far: Fix a point x in S. Let Un V_n (union of V_n's...) be an open covering of S, where V_n=B(x;n). We know that there is a...

Calculate max and min value of the function $$f(x,y)=x^2+y^2-2x-4y+8$$ in the range defined by the $$x^2+y^2≤9$$ Progress: $$f_x(x,y)=2x-2$$ $$f_y(x.y)=2y-4$$ So I get $$x=1$$ and $$y=2$$ We got one end point that I don't know what to do with $$x^2+y^2≤9$$ If I got this right it should be a...

I searched for this but couldn't find a sol. when entering the code for sets i.e. \mathbb{N} I get this error message: ! Undefined control sequence. <recently read> \mathbb l.32 $\mathbb{N}$ The control sequence at the end of the top line of your error message was never \def'ed. If...

Author: E. Kamke Title: Theory of Sets Amazon Link: https://www.amazon.com/dp/0486601412/?tag=pfamazon01-20

Homework Statement Prove that if f is uniformly continuous on a bounded set S, then f is a bounded function on S.Homework Equations Uniform continuity: For all e>0, there exist d>0 s.t for all x,y in S |x-y| implies |f(x)-f(y)| The Attempt at a Solution Every time my book has covered a...

I'm seeing the term "measurable sets" used in the definition of some concepts. But when comparing with other concepts that rely on "closed sets", I can't seem to easily find whether measureable sets are open or closed. Does anyone have any insight into that? Thanks.

Homework Statement I am confused with sets- just wanted some clarification. Say, I have a set A={b, {1,a},{3}, {{1,3}}, 3} What are the elements of set A? What are the subsets of set A? Are the subsets also the elements of the set A? The Attempt at a Solution I think the elements of the...

Homework Statement Are the following sets subspaces of R3? The set of all vectors of the form (a,b,c), where 1. a + b + c = 0 2. ab = 0 3. ab = ac Homework Equations Each is its own condition. 1, 2 and 3 do not all apply simultaneously - they're each a separate question. The...

My textbook says that "a chart or coordinate system consists of a subset U of a set M, along with a one-to-one map \phi :U\rightarrow\mathbf{R}^n, such that the image \phi(U) is open in \mathbf{R}^n." What's the motivation for demanding that the image of U under \phi be open?

(Hey guys and gals!) Homework Statement Given a bounded set x_n and for any y_n the following condition holds: \limsup_{n \rightarrow ∞}(x_n+y_n) = \limsup(x_n)+\limsup(y_n) Show that x_n converges. Homework Equations Definition of limsup(x_n) = L: \forall \epsilon > 0 \mid...

Homework Statement For simplicity, I'm leaving out extraneous details (like actual numbers). Also, apologies for my formatting; I don't know how to use Latex, but I tried to make this as readable as possible. I have a set of N measurements for τ which each have their own standard deviations...

Hello. I wish to prove this: $$\text{A function } f: X \to Y \text{ is continuous if and only if the inverse image of any closed set is closed.}$$ Proof: $(\implies)$ Let $V \subset Y$ be a closed se. By definition, $Y-V$ is an open set, and by the continuity of $f$ it follows that...

Homework Statement So this question arose out of a question about showing that a set χ is dense in γ a B* space with norm ||.||, but I think I can safely jump to where my question arises. I think I was able to solve the problem in another way, but one approach I tried came to this crux and I...

1. Among the following sets, identify all pairs of equal sets? What is the cardinality of each one of the sets? a) ∅ b) {∅} c) {{∅}} d) {∅,{∅}} e) {∅} \bigcap {{∅}} f) {{∅},∅} I would truly appreciate if you explain a bit. Thank you in advance. _____________________________________________ my...