Sets Definition and 1000 Threads

I was just reading an article the other day about the debate in public schools about teaching evolution as an absolute truth as it has been taght for the past umteen years. Not saying that I'm a proponent of creationism or even that I'm not, but there are some serious flaws in teaching it as the...

Im not sure how to interpret the notation, specifically the one on the left, the one on the right seems like you just include everything in the universal set? what does it mean when the line goes over everything? what does it even mean when the line is over the and/or symbol

For a pair (A,B) of subsets of the set X=(1,2,...100), let A*B denote the set of all elements of X which belong to exactly one of A or B. what is number of pairs (A,B) of subsets of X such that A*B=(2,4,6,...100)? I let A =(1,2,3...50) and B=(51,52,...100) so there are 25 elememnts of...

Homework Statement Given that F is a family of sets, that \bigcup F is the union of the sets members of the family F, that A is a set, assume that (1) \hspace{1cm} \forall F (\bigcup F = A \rightarrow A \in F) then prove that (G) \hspace{1cm} \exists x (A= \left\{ x \right\} )...

Homework Statement Show that the collection \{ \{a\}\times(b,c) \subset \mathbb{R^2} |a,b,c \in \mathbb{R} \} of vertical intervals in the plane is a basis for a topology on \mathbb{R^2} The Attempt at a Solution My question is just really about (a)X(b,c) am I just basically...

Homework Statement Let M be a metric space with metric d, and let d_{1} be the metric defined below. Show that the two metric spaces (M,d), (M,d_{1}) have the same open sets. Homework Equations d_1:\frac{d(x,y)}{1+d(x,y)} The Attempt at a Solution I tried to show that the neighborhoods...

Homework Statement Hi I would like you please to look my attachement ,and explain to me the meaning of the line above M and N what's the meaning of this line?It seems to me that it acts as we should take the complementary collection of numbers . Homework Equations The Attempt at a...

Homework Statement Hello. Here is the question: Determine whether or not R is some sort of order relation on the given set X. X = {∅, {∅}, {{∅}} } and R ε ⊆. I can't seem to figure out why the ordered pairs given are what they are. Homework Equations None. The Attempt at...

Hey all, I was reading Terence Tao's text on analysis. After stating the axioms of pairing and regularity, he asks for proof of the statement that no set can be an element of itself, using the above two axioms. He has not defined any concepts like hierarchy or ranks. I can see how, A...

Homework Statement Let \displaystyle f:{{\mathbb{R}}^{n}}\to \mathbb{R} a continuous function. Proove that: If \displaystyle f\left( p \right)>0 then there's a ball \displaystyle {{B}_{p}} centered at p such that \displaystyle \forall x\in {{B}_{p}} we have \displaystyle f\left( x...

Hi, sorry if this is in the wrong section. Some of the stuff in this section is way over my head anyway. I have 10 sets of 3 numbers ranging from 1 to 10. They are interesting in that each number appears three time, no number appears twice in the same set, and no two numbers appear together in...

Hello, I am trying to teach myself set theory...main problem is, as an engineer, mathematical proofs were never exactly stressed in my curriculum. (Scary, right?) The problem is stated as follows: "Prove the following, {x\inZ|for an integer y, x=6y}={x\inZ|for integers u and v, x=2u...

Lets say I have \aleph_1 numbers of sets that each have \aleph_1 number of elements and I want to show that the union of all of these sets has \aleph_1 number of elements. I start with the line segment [0,1] and shift this line segment up by all the reals from 0 to 1. So now...

I am having some doubts in the definitions of the upper and lower integrals in apostol. There is a statement saying "Let S denote the set of all numbers _{a}\int ^{b} s(x) dx obtained as s runs through all step functions below f i.e. S = { _{a}\int ^{b} s(x) dx | s < f} " I did not get...

Homework Statement Suppose that T:W -> W is a linear transformation such that Tm+1 = 0 but Tm ≠ 0. Suppose that {w1, ... , wp} is basis for Tm(W) and Tm(uk) = wk, for 1 ≤ k ≤ p. Prove that {Ti(uk) : 0 ≤ i ≤ m, 1 ≤ j ≤ p} is a linearly independent set.Homework Equations The Attempt at a Solution...

Hi, All: I am trying to find a construction of a measurable subset that is not Borel, and ask for a ref. in this argument ( see the ***) used to show the existence of such sets: i) Every set of outer measure 0 is measurable, since: 0=m* (S)≥m*(S) , forcing equality. ii) Every...

Homework Statement http://i.imgur.com/tjpka.png (the actual problem is the third part down) Homework Equations the first two parts are the definition of borel sets,and the second part is a relevant theorem. The Attempt at a Solution so I'm new to Borel sets. And I feel like I'm...

Does this mean that whenever the function P(x) is true, then x is an element of X, and when P(x) is false, then x is not an element of X? I'm confused because the wording says that "...a sentence P(x) that is either true or false whenever x is any particular element of X..." which leads me to...

I apologize for the repost, but I had no replies to my previous post. I figured that I didn't put down a good enough attempt of a solution. I will try to explain what I did in more detail. I have read the rules for the forum, but if I'm still doing something wrong, please tell me. I want to...

As the post title describes, I have two data sets and want to multiply them together (and finally plot them). The problem I have is that the data are different length, so for example using Matlab: >>> data_set1 .* data_set2 But this would not give me the correct answer for a number of...

Hello, everyone. I've trouble attempting to read the following. One of the things we assume for the set of real numbers is. A map $\left(\xi, \eta\right) \to \xi+\eta$ from $\mathbb{R} \times \mathbb{R}$ into $\mathbb{R}.$ Could someone read the above in plain English, please. Does it mean all...

why intersection of empty class of sets is the whole space while their union is null set? Book writes that an element will fail to be in the intersection if it fails to be in one of the sets of the class but since there is nothing in the empty class so there is nothing in the empty class that...

say i have 500,000 0s or 1s. say i have 50 such sets, each that i have ranked or assigned a value to - "harshness". can i then extrapolate - is that the right word - to find the perfect dataset that instantiates the property of harshness? and can i measure the harshness of other datasets...

Let A={1, 2} and B={∅}. First, I find the power set of A and the power set of B: P(A)= { ∅, {1}, {2}, {1, 2} } P(B)= { ∅, {∅} } I believe the power sets are correct. I'm still new to the concept of power sets. Anyway, my main question is regarding cartesian product of power sets. I'm...

A question about sets?? I have a number of weird shaped flat objects. I am interested in covering as much of the floor as I can. After placing the objects on the floor, the only info I have is: Choosing any two objects on the floor, the overlap between them is at a minimum possible...

Hi, I was reading over a solution after working on a problem and got confused about some parts: http://nweb.math.berkeley.edu/sites/default/files/pages/f10solutions.pdf (first problem) First, how do we know that there are disjoint open sets U and V for each of the separated sets? (does...

Calculating "match" between two data sets Hey guys, I'm developing a program for comparing the effects of various terms in a Monte Carlo experiment. Right now I have it so you can visually see the effect of "switching" terms on and off and need a way of quantifying how much two lines "match"...

[Note: If this is posted in the wrong forum, I'm very sorry. It is directly related to a textbook question.] This may be a silly question. I know that I can prove two sets to be equal by showing that they are subsets of each other. But, what if I have that two sets have the same...

In the expression of sets: B={X \in A:|X|<3} the expression is saying that B is a set that contains at most 3 sets X that belongs to A, right? How do we say, B is a set that contains elements of X that belongs to A, and all X elements contains at most 3 x elements (the cardinality of X is at...

for example i have this : F={1,2,3,4,5} so F=1,2,3,4,5 but how to randomize the set ? i want to say F=5,3,4,2,1 or 2,3,1,4,5 or ... do i have to say like this? : F=(1)/(2)/(3)/(4)/(5)

I read that an empty collection of sets, denote it by λ, is a little problematic when one considers \bigcup_{A\inλ}A and \cap_{A\inλ}A. I can see that the union should be ∅. However, for the intersection it was argued that if one considers a set X to be the universe of the discussion then the...

Homework Statement Let A be a set. Show that there is no surjective function phi: A --> P(A), where P(A) is the power set of A. What does this say about the cardinalities of A and P(A)? Homework Equations Assume that phi is a surjection of A onto P(A) and consider the set U= {a in A : a...

Homework Statement Suppose that (X,\tau) is the co-finite topological space on X. I : Suppose A is a finite subset of X, show that (A,\tau) is discrete topological space on A. II : Suppose A is an infinite subset of X, show that (A,\tau) inherits co-finite topology from (X,\tau). The...

Homework Statement Show that for two well ordered sets, (A, R) and (B, S), the disjoint union of A and B will be well ordered by the relation R \cup S \cup A \times B . The Attempt at a Solution ... I honesly don't know how to start at this one..

I continue with a questions regarding proofs in set theory.. :) Halmos just writes that every ordinal number is a transitive set but doesn't prove it. Is there any simple proof of this?

Hello everyone! Welcome to the inaugural POTW for Graduate Students. My purpose for setting this up is to get some of our more advanced members to participate in our POTWs (I didn't want them to feel like they were left out or anything like that (Smile)). As with the POTWs for the...

(a u b)'=a' u b'

Regular open sets,,,, If U is an open set in a topological space (X,τ),is it true that U=〖int〗_X 〖cl〗_X U?Justify.

Hello All, I am finding the hardest time in understanding how to work δ & ε Open Set Problems? Can someone please explain this approach to me? Thanks in Advance

is it possible to make a venn diagram wherein the elements are infinite elements? ex. V = { is the set of all odd numbers) W = { 5, 15, 25, 45, 55,...} thanks a lot

W = { x| 0< x < 3} Y = { x| x > 2 } Z = { x | 0 <= x < = 4} then the problems: 1. (WUY) intersects Z = 2. (W intersects Y) intersects Z = do my propose answers below correct sir/mam? 1. 0 < x < = 4 2. 2< x < 3 hope you can help me on this im using the line number ... but all i see in the...

continuous -- how can I combine these open sets Homework Statement let ##X,Y## be compact spaces if ##f \in C(X \times Y)## and ## \epsilon > 0## then ## \exists g_1,\dots , g_n \in C(X) ## and ## h_1, \dots , h_n \in C(Y) ## such that ##|f(x,y)- \Sigma _{k=1}^n g_k(x)h_k(y)| < \epsilon...

I am reading Martin Crossley's book - Essential Topology - basically to get an understanding of Topology and then to build a knowledge of Algebraic Topology! (That is the aim, anyway!) On page 27, Example 3.33 (see attachment) Crossley is explaining the toplogising of \mathbb{R} P^2 where...

Homework Statement I was given a pdf document containing questions that require me to prove set rules. However, the third question (the one that starts at the bottom of the first page and runs into the second page) is giving me problems. I might be able to prove it if he wants a proof by...

Hi all. In chapter 9 of Halmos's book titled Naive set theory, he talks about families of sets. He then talks about the associativity of sets as follows "The algebraic laws satisfied by the operation of union for pairs can be generalized to arbitrary unions. Suppose, for instance, that {Ij}...

Question is in paint doc. Determine if the statement is true or false. My solution: I have two solutions Sol 1: FalseIf A1 contains A2 and A2 contains A3 then the number of elements of A3 contained in A1 is less than the number of elements in A2 contained in A1. In other words the...

Homework Statement It's given or I've already shown in previous parts of the question: A \in M_{nxn}(F)\\ A^{2}=I_{n}\\ F = \mathbb{Q}, \mathbb{R} or \mathbb{C}\\ ker(L_{I_{n}+A})=E_{-1}(A) Eigenvalues of A must be \pm1 Show im(L_{I_{n}+A})=E_{1}(A) where E is the eigenspace for the eigenvalue...

Just a quick question. If Q is a dense set of a metric space X, and P is a dense set of a metric space Y, then is Q x P a dense set of X x Y? I am fairly sure this is the case. If this is true, then I want to use this statement to show that the open sets of the product of finite number of...

This question is in regards to higher dimensional algebraic geometry. The actual problem is very complicated so here is my question which is substantially simplified. Suppose {f_1,... f_k} is a set of quadratic polynomials and {g_1,...,g_l} is a set of linear polynomials in a polynomial ring...

Homework Statement Decide wheter the following sets are dense in ℝ, nowwhere dense in ℝ , or somewhere in between. a) A= \mathbb{Q} \bigcap [0,5] b) B= \{ \frac{1}{n} : n \in \mathbb{N} d) the cantor set. The Attempt at a Solution a) so we have the rationals intersected with...