Set Definition and 1000 Threads

Hello! (Wave)I want to show that the algebraic set of $K^n$, $V$, is irreducible iff $I(V)$ is a prime ideal. That's what I have tried so far: We know that the algebraic set $V$ is irreducible iff $V$ cannot be written as $V=V_1 \cup V_2$, where $V_1, V_2$ are algebraic sets of $K^n$ and $V_1...

Prove that for all $$x,y\in\omega,\ \ x\subset y\vee y\subset x.$$ If I assume that the conclusion is false then I can prove that for some $$a\in x,\ b\in y$$ we have $$a\notin b$$ and $$b\notin a.$$ Also I am thinking that if assume the contrary then $$\omega$$ minus $$\{x\}$$ or minus...

Hey guys, I was reading Kenneth's Discrete Mathematics and I came across this definition in the function chapter: Let f be a function from A to B and let S be a subset of A.The image of S under the function f is the subset of B that consists of the images of the elements of S.We denote...

Let $$x$$ be a natural number (set). How to prove that there is no bijection between $$x$$ and $$x^+$$, where $$x^+=x\cup\{x\}$$? Then I can show that $$\mathrm{card}\,x<\mathrm{card}\,x^+.$$ I know that $$x\notin x.$$

Hi! (Smile) Let $B$ be a nonempty set. Does it stand that $\bigcap \mathcal{P}B=\mathcal{P} \bigcap B$? Is the set $B \times B$ always a function? If not, what condition should $B$ satisfy, so that the relation $B \times B$ is a function? Let $x \in \bigcap \mathcal{P}B$. Then $\forall b \in...

Hey! (Wave) If $x \neq y $, define the sets $\bigcup \langle x,y \rangle , \bigcup \bigcup \langle x,y \rangle$. According to my notes, it is like that: $$\langle x,y \rangle= \{ \{x\}, \{x,y\} \} $$ $$ \bigcup \langle x,y \rangle=\{x,y\} $$ $$ \bigcup \bigcup \langle x,y \rangle=x \cup y$$...

Hello! (Wave) Let $R$ be a relation and $A$ a set. The restriction of $R$ to $A$ is the set: $$R\restriction A=\{ <x,y>: x \in A \wedge <x,y> \in R\}=\{ <x,y>: x \in A \wedge xRy\}$$ For a relation $R$ and a set $A$, it stands that: $$dom(R \restriction A)=dom(R) \cap A$$ Could you give...

ℝ or ℝ2? Both are infinite, but is one greater then the other?

Homework Statement Hi there! I have a data set of r (independent variable) and E (electric field strength) (dependent variable). The question asks for a non graphical method to show if there is an inverse square law relationship between the two data sets. -- My attempt: I picked the equation...

Hi guys, I stumbled upon this lovely quote from the philosopher of science Wesley Salmon: "The fool hath said in his heart that there is no null set. But if that were so, then the set of all such sets would be empty, and hence, it would be the null set. Q.E.D." (in Martin Gardner, Mathematical...

Homework Statement Prove if an ordered set A has the least upper bound property, then it has the greatest lower bound property. Homework Equations Definition of the least upper bound property and greatest lower bound property, set theory. The Attempt at a Solution Ok, I think that my main...

Homework Statement Let ##G## be a group. I need to find a set ##X## and an injective function from ##G## into ##Sym(X)## Homework EquationsThe Attempt at a Solution I am having difficulty with this problem, and I want to make sure I understand exactly what it is asking. If I understand the...

If you take the Fourier series of a function $f(x)$ where $0 < x < \pi$, then would $a_{0}$, $a_{n}$, and $b_{n}$ be defined as, $a_{0} = \displaystyle\frac{1}{\pi}\int_{0}^{\pi}f(x)dx$ $a_{n} = \displaystyle\frac{2}{\pi}\int_{0}^{\pi}f(x)\cos(nx)dx$ $b_{n} =...

Homework Statement Determine whether the set S spans R3. If the set does not span R3, then give a geometric description of the subspace that it does span S = [ (2,0,3) , (2,0,-1) , (6,0,5) , (4,0,6) ] Homework EquationsThe Attempt at a Solution I know S does not span R3 because the system of...

Hello everyone, Here is the definition of path connected and domain my textbook provides: Definition 171. An open set S is path connected if each pair of points in S can be connected by a polygonal line (e.g. a finite number of line segments connected end to end). A domain is an open set that...

Could anyone recommend some good books on set theory?

Mod note: Moved from a technical section, so missing the homework template. Here is what I'm trying to prove. Let f:A->B. If there are two functions g:B->A and h:B->A such that g(f(a))=a for every a in A and f(h(b))=b for every b in B, then f is bijective and g=h=f^(-1). I think I have most...

Hello, I have the following problem. I have a system of differential equations, with two parameters that satisfy certain condition. 0 < 1.5(1-a) < b < 1. So when I fix the value of a I can find values of b satisfying this and its associated equilibrium point. When I calculate (with...

Image is a set of 1D potentials which i need more examples and their solutions containing transmitting states, bounded states, scattering states and coefficients. I searched with "1D potential combinations" "1D potential set" keywords but can not find anything yet. Which keyword should i...

My question is on how to answer if two statements are equal in set theory. Like De'Morgans laws for example. I'm currently reading James Munkres' book "Topology" and am working through the set theory chapters now, and this isn't the first time I've seen the material, but every time I see this...

Homework Statement Determine if the given set of functions is linearly independent or linearly dependent.Homework Equations $$S=x~sin~x, ~ x~cos~x$$The Attempt at a Solution My first instinct was to use the Wronskian. $$W[y_1(x), y_2(x)]=\begin{vmatrix} x~sin~x & x~cos~x\\ x~cos~x+sin~x &...

Let B be a non-zero mx1 matrix, and let A be an mxn matrix. Show that the set of solutions to the system AX=B is not a vector space. I am thinking that I need to show that the solution is not consistent. In order to do so would I need to show that B is not in the column space of A?

Homework Statement Let S = { (x,y) in Z | 5x+7y is divisible by 3 } Show that S is symmetrical. Homework Equations None apart from basic algebraic knowledge. The Attempt at a Solution [/B] The only thing I can think of is starting with 3a = 5x+7y and putting x (or y) into the...

Homework Statement Damped driven oscillator: ruler example. Suppose the ruler used in the classroom demonstration has a length of 12 and 13/16ths inches, a width of 1 ½ inches, is 1/16th inch thick with a density of 1.2 g/cm3. It swings from a pivot point ¼ of an inch from the top end. a) Find...

Hi guys, I've got a philosophical question for you concerning the empty set. I know that in axiomatized set theory the empty set is disjoint with itself. Because it has no members, the empty set cannot have any members in common with itself. This is common sense in set theory. But if we say...

Hi everyone. It's been years since I've done any stats, so I need a bit of help, please. I want to include it in a blog post I'm going to do (not here on PF), so I don't want to give away too many details :p I apologise for my terrible understanding of stats, please be patient! Anyway, over...

Homework Statement This is from Baby Rudin Exercise 2.20- Are closures and interiors of connected sets always connected? (Look at subsets of \mathbb{R}^2 ). Homework Equations The interior is the set of all interior points for a set E that is a subset of a metric space X. A subset Y of a...

Hello! Why does the moon has different rise and set timing for different phases? And also, why does it rise from different directions during different phases? Help me out please!

Find the solution set for x^2/(x+3) < 9/(x+3) So I moved the term 9/(x+3) over to the left side and cross-multiplied the two fractions. Then, I simplified to get x^2-9 (because the x+3 cancel out across the fraction bar). I got x^2-9, which factors to (x+3)(x-3). Then, I created an interval...

Hey! (Wave) Knowing that $A,B$ are sets, and: $$\text{ If } a \in A, \text{ then } \{ a \} \subset A \rightarrow \{ a \} \in \mathcal P A \rightarrow \{ a \} \in \mathcal P (A \cup B)$$ from this: $\{ a \} \in \mathcal P (A \cup B)$, can we conclude that: $$\{ a \} \subset \mathcal P (A \cup...

Hi! (Smile) I want to prove that for each set $A$: $$A \subset \mathcal P \cup A$$ According to my notes, we prove it like that: Let $x \in A$. We want to show that $x \in \mathcal P \cup A$, so, that: $\exists y \in \mathcal P \cup A$, such that $x=y$. It suffices to show that if $z \in x$...

There is a Theorem that states that if G = (V, E) is a graph, then S is an independent set $\Leftarrow\Rightarrow$ V - S is a vertex cover. Suppose the vertices have positive integer weights. Does it follow from the theorem that: S is an independent set with maximum weight...

Homework Statement Suppose that R is a partial order on A, B1 ⊆ A, B2 ⊆ A, x1 is the least upper bound of B1, and x2 is the least upper bound of B2. Prove that if B1 ⊆ B2 then x1Rx2. Homework EquationsThe Attempt at a Solution I split the proof into two different cases: case 1: x_1 is an...

In Chapter 1: "Integral Domains", of Saban Alaca and Kenneth S. Williams' (A&W) book "Introductory Algebraic Number Theory", the set of all Eisenstein integers, $$\mathbb{Z} + \mathbb{Z} \omega$$ is defined as follows:https://www.physicsforums.com/attachments/3392Then, Exercise 2 on page 23 of...

Question: Show that the set of all functions of the form f(x) = ax+b, with a and b real numbers forms a vector space, but that the same set of functions with a > 2 does not. Equations: the axioms for vector spaces Attempt: I think that the axiom about the zero vector is the one I need to use...

Hello, I know that the law of the excluded middle is implied in ZFC set theory, since it is implied by the axiom of choice. Taking away the axiom of choice, does ZF set theory (with axioms as stated in the Wikipedia article http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory)...

In Paul Bland's book: Rings and Their Modules, we read the following text at the start of Section 2.2 Free Modules:https://www.physicsforums.com/attachments/3388In the above text we read: " ... ... Every submodule has at least one set of generators, namely the set $$N$$. ... " Now, I know that...

Homework Statement A spherical conductor of radius ##a## carries a charge q and also there is a jelly of constant charge ##rho## per unit volume extending from radius ##a## out to radius ##b##. I'm looking to see if I got the correct set up for the electric field of this spherical conductor for...

Homework Statement In the linear space of all real polynomials with inner product (x, y) = integral (0 to 1)(x(t)y(t))dt, let xn(t) = tn for n = 0, 1, 2,... Prove that the functions y0(t) = 1, y1(t) = sqrt(3)(2t-1), and y2 = sqrt(5)(6t2-6t+1) form an orthonormal set spanning the same subspace...

Background: I am trying to write a program for Buckingham Pi Groups. I need to find a way to list all the input varialbes as different sets. For example if I have 4 variables [V D p u] and I want to distribute them 3 ways I get 4 sets. Number of Sets = Binomial(Number of Variables...

Hey there guys, its AngrySnorlax here again with another problem. I posted here before when I was having an issue and the responses I got were extremely helpful because there was a specific step that I just could not grasp that was explained to me and I am hoping that is the same situation here...

Hey! (Wave) Theorem (Russell's paradox is not a paradox in axiomatic set theory) The set of all sets does not exist. Proof We suppose that the set of all sets exist, let $V$. So, for each set $x$, $x \in V$. We define the type $\phi: \text{ a set does not belong to itself, so } x \notin x$...

Hi! (Cool) I want to show that $\cap \varnothing$ is not a set. That's what I have tried so far: We suppose that $\cap \varnothing$ is a set. Let $x \in \cap \varnothing$. Then, $\forall b \in \varnothing, x \in b$. However, $\varnothing$ does not contain any element. So, we cannot find a $b...

What's the set \{ z \in \mathbb{C}| |z|^2 \geq z+ \bar{z} \}? I've set z=a+ib and found a^2 + b^2 \geq 2a \Rightarrow b^2 \geq a(2-a) I'm not sure how to interpret this geometrically ie what it looks like? I suppose it is the set of vectors whose length is bigger than twice their real part. I...

Hello! (Wave) I want to show that $A \subset \varnothing \rightarrow A=\varnothing$. That's what I thought: $$A \subset \varnothing \text{ means that :}$$ $$\forall x (x \in A \rightarrow x \in \varnothing)$$ Since, there is no $x$, such that $x \in \varnothing$, there is no $x$, such that...

In all the topology textbooks I used in school, the open set axoims specified 4 conditions on a set S: (i) S is open (ii) empty set is open (iii) arbitrary union of open sets is open (iv) finite intersection of open sets is openI noticed on proofwiki, that (ii) is omitted. I was curious if...

Hey again! (Blush) I want to show that the set $\{ a, b \}$ is unique.That's what I have tried: We suppose that $\{a,b\}, \ \{a,b \}'$ are sets, so that each of them has as elements $a$ and $b$ and only these ,and $\{a,b \} \neq \{a,b \}'$. From the axiom of extensionality, there is, without...

South Africa is set to get a nuclear power plant, following the signing of a cooperation deal with Russia recently. Both sides noted that the nuclear power plant will have a production capacity of up to 9.6 GW based on Russian Technology, by 2030. Read more here...

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.

Hello! (Wave) What is the subject Set Theory about? What knowledge is required? (Thinking) That is the Course Content: Brief report on basic elements (algebra of sets, relations and functions, etc..). Construction of the set of natural numbers. Ordinal numbers and their arithmetic. The...