Set Definition and 1000 Threads

Homework Statement Show that if a set has 3 elements, then we can find 8 relations on A that all have the same symmetric closure. Homework Equations Symmetric closure ##R^* = R \cup R^{-1} ## The Attempt at a Solution If the symmetric closures of n relations are the same then...

Homework Statement Hey guys! I am new to this forum but saw the helpful posts on set theory proofs and wondered if I could finally get some help with this problem: Suppose A, B, C, and D are sets with A⊆C and B⊆D. If A∩B=Ø then C∩D=Ø. This is a biconditional so I have to prove it...

Hey guys! I am new to this forum but saw the helpful posts on set theory proofs and wondered if I could finally get some help with this problem: Suppose A, B, C, and D are sets with A⊆C and B⊆D. If A∩B=Ø then C∩D=Ø. This is a biconditional so I have to prove it both ways correct...

Homework Statement Use polar coordinates to set up and evaluate the double integral f(x,y) = e-(x2+y2)/2 R: x2+y2≤25, x≥0 The Attempt at a Solution First I just want to make sure I'm understanding this my double integral would be ∫^{\pi/2}_{-\pi/2} because x≥0 ∫^{5}_{0}...

Homework Statement For a reaction, A + H2O --> B + C We're given that d[A]/dt = k[A]n[H3O+]m And also a table of [A] vs time at T1 and pH 1, pH 2; as well as [A] vs time at T2 and the same pH 1 and 2. From this data, we're to find pseudo-n-order rate constants, and then n itself. Next...

Am a bit confused about the meaning of cardinality. If ## A \subseteq B ##, then is it necessarily the case that ## |A| \leq |B| ##? I am thinking that since ## A \subseteq B ##, an injection from A to B exists, hence its cardinality cannot be greater than that of B? But this cannot be...

Homework Statement . Let ##X## be a complete metric space and consider ##C(X)## the space of continuous functions from ##X## to ##\mathbb R## with the metric ##d_{\infty}##. Suppose that for every ##x \in X##, the set ##\{f(x): f \in C(X)\}## is bounded in ##\mathbb R##. Prove that there exist...

##GL(n,\mathbb{R})## is set of invertible matrices with real entries. We know that SO(n,\mathbb{R}) \subset O(n,\mathbb{R}) \subset GL(n,\mathbb{R}) is there any specific subgroups of ##GL(n,\mathbb{R})## that is highly important.

**Let A and B be two subsets of some universal set. Prove that if $(A\cup B)^c$ = $A^c$ U $B^c$, then A = B.**Attempt: Let $x\in A$. Then $x\in A\cup B$, so $x\notin(A\cup B)^c$. By hypothesis $(A\cup B)^c=A^c\cup B^c$, so $x\notin A^c\cup B^c$. In particular, then, $x\notin B^c$, and therefore...

Hi all, first math post here. I was just wondering- after having read from quite a few textbooks that intuitively, a set is a collection of objects- if there's a rigorous definition of the concept of set. It's just out of curiosity- I mean, is a rigorous definition even necessary? I guess I'm...

I am currently reading Rautenberg's book on mathematical logic, in it he defines a propositional language ##\mathcal{F}##, set theoretically, as the smallest (i.e. the intersection) of all sets of strings ##S## built from propositional variables (##\ p_1,p_2,\ldots##) as well as any binary...

Homework Statement Prove that(power set) P(E) U P(F) is a subset of P(E U F) Homework Equations P(E) U P(F) is a subset of P(E U F) The Attempt at a Solution P(E)U P(F)={x:xεP(E) or xεP(F)} but P(E)={X:X is a subset of E} or P(E)={x:xεX→xεE} so we get P(E)U P(F)={x:xεX→xεE or...

I was thinking of finding a new hobby, and thought that playing around with the double slit experiment might be interesting. I was wondering how feasible it would be to set up the double slit experiment, not simply the laser and slit version shown here...

Homework Statement In a group of 30 people each person twice read a book from books A, B, C. 23 people read book A, 12 read book B and 23 read book C. (a) How many people read books A and B? (b) How many people read books A and C? (c) How many people read books B and C? Homework...

Homework Statement A set of functions, F, is given below. Determine the size of the largest subset of F which is mutually orthogonal on the interval [-1, 1], and find all such subsets of this size. Show all of your work. F = { 1, x, x2 , sin(x), cos(x), cosh(x), sinh(x)}Homework Equations Not...

Homework Statement . Let ##X## be a compact metric space. Prove that if ##\mathcal F \subset X## is a family of equicontinuous functions ##f:X \to Y \implies \mathcal F## is uniformly equicontinuous. The attempt at a solution. What I want to prove is that given ##\epsilon>0## there...

The topology ## T ## on a set ## X ## generated by a basis ## B ## is defined as: T=\{U\subset X:\forall\ x\in U\ there\ is\ a\ \beta\in B:x\in \beta \ and\ \beta\subset U \}. But if ##U## is the empty set, and there has to be a ## \beta ## in ##B## that is contained in ##U##, the empty set...

[FONT=arial]Prove that a set $A\subset\mathbb{R}^n$ is (Lebesgue) measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$ such that $B\subset A\subset C$ and $C$~$B$ (C without B) is a null set. $F_{\sigma}$ is a countable union of closed sets, and...

Prove that the interval A = [0 , 2) has the same cardinality as the set B = [5 , 6) U [7 , 8) by constructing a bijection between the two sets Attempt: x ↦ x + 5 for x ∈ [0 ; 1) x ↦ x + 6 for x ∈ [1 ; 2) What to do next?

Algebraically, how is this done? I can do it no problem if there is no fraction, but have problems when there is. (1/n) = (n/100)

1. Prove that if A is symmetric and B is skew-symmetric, then {A,B} is a linearly independent set. I am going to need some help to solve this. Not sure how to begin. Homework Equations The Attempt at a Solution

How do you show that a set of linear transformations from one vector space to another spans L(V,W)? This isn't a homework question, or even a question that's in the text I'm reading (Friedberg).

I have been looking at the idea of 1:1 correspondence as a method of determining set size/cardinality, and have noticed that the principle allows for inductive proofs, which I think are properly constructed, that can come to conclusions which are clearly wrong under traditional set theory if...

How to show that a set ##C=\{(x,y,z)\in\mathbb{R}^3:x\geq0,z\geq0,xz\geq y^2\}## is convex? I tried a proof by contradiction: Assume that there exist ##c_1=(x_1,y_1,z_1),c_2=(x_2,y_2,z_2)\in C## and ##t\in(0,1)## such that ##tc_1+(1-t)c_2\notin C##. For this to hold, one would have to have...

Homework Statement Explain why ##f(x,y,z) = x + y - z## must attain both a maximum and a minimum on the sphere ##x^2 + y^2 + z^2 = 81)##. Homework Equations None The Attempt at a Solution I know that any continuous function attains both a maximum and a minimum on a compact set. I defined...

Hi Please see table below GDP growth rates of a group 5 countries. I am trying to derive some conclusions from this small sample. 1. can I conclude that countries in the group have very similar growth rates and there is no significant difference between their growth rates? 2. As...

Homework Statement Give f:A→A and g:A→A where f is surjective, g is injective, but f*g is neither surjective nor injecive The Attempt at a Solution I don't know why I can't really think of two... I assume it's easiest to do one in ℝ, but when it comes to producing...

how do I go about doing 3(a) and 3(b)? I'm guessing that for 3(a), it is true, since we have for LHS: P((A or B) and C) we can consider the case P(A and C) by excluding B, and this is a subset of the RHS when we also exclude B: (P(A) and P(C)). We can consider excluding B because...

Homework Statement 2.) if jimmy piles his baseball cards in stacks of 4, then there is 1 left over. if he piles them in stacks of 7, there are 4 left over. If he piles them in stacks of 9, there are 6 lefty over. If he piles them in stacks of 10, there are 7 left over. compute the smallest...

Hi, i would like to have a hint for the following problem: Let $$v_1, v_2 \&\ v_3 $$ in a vector space V over a field F such that$$ v_1+v_2+v_3=0$$, Show that $\{v_1,v_2\}$ spans the same subspace as $\{v_2,v_3\}$ Thanks in advance

Graph Theory -- Max. Independent set algorithm Homework Statement Design a polynomial time greedy algorithm to compute a maximum independent set for a graph. Explain the algorithm and compute T_w(n). Homework Equations The Attempt at a Solution My terse and informal...

How would I go about proving that if A is a subset of B then the interior points of A are a subset of the interior points of B?

Hello all, I am struggling with this relatively simple task. In a university with 88 students, each student can choose to participate in 3 afternoon activities: activity A, activity B and activity C. Each student can choose to participate in some activities, all or none. 33 students...

Homework Statement Let X be a set containing n elements. If two subsets A and B of X are picked at random, the probability that A and B have the same number of elements is Homework Equations The Attempt at a Solution Total number of subsets possible is 2^n. Now the subsets containing 1...

hi Guys, i Needed your help to prove out the following, thanks in advance; Let u1,u2,...,ut be vectors in $\Re^n$ and $k\in\Re ,k\neq0.$ Prove that $Span\{u_1,u_2,...,u_t\}=Span\{ku_1,u_2,...,u_t\}$

Hi everyone, let's stay I have two inequation set such as: First one is A:= $$X_1-X_2 \leq 1$$ $$X_1 \leq3$$ $$X_2 \geq 1$$ $$X_1,X_2 \geq 0$$ Second one is B:= $$X_1+X_2 \geq 5$$ $$X_1\leq5$$ $$X_1\geq4$$ $$X_2\leq4$$ $$X_1,X_2 \geq 0$$ I had like to write it as a set $$C := A\oplus B$$...

Homework Statement Benzoic acid/biphenyl mixture is mixed with diethyl ethe(CH3CH2-O-CH2CH3). The solution is placed in a separatory funnel and sodium hydroxide is mixed in. The bottom layer, H2O, NaOH and benzoic acid is then placed in a flask where it is mixed with HCl, chilled and then...

Hi everyone, :) I encountered the following question recently. :) Now I think this question is wrong. Let me give a counterexample. Take the set of real numbers with the usual Euclidean metric. Then take for example the sequence, \(\{\frac{1}{n}\}_{n=1}^{\infty}\). Then...

Homework Statement Homework Equations The Attempt at a Solution I'm not sure how to prove that it is zero. I don't see what I can do after the second last step.

Homework Statement Let T be a set such that: T=\{t\in\mathbb{R}/t^{2}<2\} Homework Equations a) Justify the existence of a real number a such that a=Sup(T) b) Prove that the proposition a^{2}<2 is false. c) Suppose that a^{2}>2. Prove that we can find a contradiction with a=Sup(T). d)...

Why is the math in the red box necessary? According to this definition, it isn't:

Homework Statement b(B) = cls(B) \ Int(B) where b(B) is the boundary, cls is the closure, and int is the interior of set B. This was not hard for me to prove by picking elements and showing that the sets were contained in one another. However, I decided it would be fun to try to derive it by...

Suppose that A is subset of R (real line) with the property for every ε > 0 there are measurable sets B and C s.t. B⊂A⊂C and m(C\B)<ε Prove A is measurable By definition A is measurable we need to prove m(E)=m(E∩A)+m(E\A) for all E the ≤ is trivial enough to show ≥: Since C is...

Suppose A is not a bounded set and m(A∩B)≤(3/4)m(B) for every B. what is m(A)?? here, m is Lebesgue Outer Measure My attemption is : Let An=A∩[-n,n], then m(A)=lim m(An)= lim m(An∩[-n,n]) ≤ lim (3/4)m([-n,n]) = infinite. is my solution right? I am confusing m(A) < infinite , it...

On the set of Z of integers define a relation by writing m \triangleright n for m, n \in Z. m\trianglerightn if m-n is divisble by k, where k is a fixed integer. Show that the quotient set under this equivalence relation is: Z/\triangleright = {[0], [1], ... [k-1]} I'm a bit new the subject...

Hi, everyone. I once saw a science program with the danish astrophycisist Jens Martin Knudsen, who said that there exists seven absolutely fundamental constants of nature, and if one of these were changed ever so slightly, it would lead to drastic changes in the whole universe. So my...

I have a problem asking to prove the following statement is false: "Every non-empty set of integers has a least element". This seems pretty intuitively false, and so I tried to sum that up in the following way: Suppose we have a subset \(A\) in the "universe" \(X\). Let \(A=\{-n: n\in{N}\}\), a...

How to prove that the limit \lim_{n\to\infty}sin(n)^n n integer towards infinity does not exist ? If n is a real then it's obvious since we can take n=Pi/2*k k being an integer. But if n is a integer then sin(n) is always smaller than 1, hence the power n should tend towards 0. I know this...

Hello, I have a data set that follows an equation similar to sin(x)+x. Just from eyeballing the data, it seems like there should be a pretty simple trigonometric function A*sin(B*x)+C*x. I went to school for engineering so I have some basic/intermediate knowledge of mathematics but it's...

Homework Statement Let S ⊆ R be such that (i) a, b ∈ S ⇒ ab, a + b ∈ S (ii) for all x ∈ R exactly one of the following holds x ∈ S, x = 0, −x ∈ S. Show that S = {x ∈ R ; x > 0} (the set of positive numbers P) 2. Relevant theorems (T1) a² > 0 ∀ a ∈ R. (So a²∈P) (T2) All positive...