Union Definition and 218 Threads

So, my approach and solution are as follows: $$ [x * y] = \{ z \in M : z \sim (x * y) \} $$ Since we know that ##a * b \sim a^{\prime} * b^{\prime}## we can rewrite ##z## as ## x^{\prime} * y^{\prime} ##. Plugging this in yields: $$ [x * y] = \{ x^{\prime}, y^{\prime} \in M : x^{\prime} *...

Is there a name for the union of {prime numbers} and {integers that are not powers of integers}? For example, we would include 2, 3, 5, 7, 11... And also 6, 10, 12... But we exclude 2^n, 3^n, ... and 6^n , 10^n , etc. What are some interesting contexts where this set crops up?

I'm watching Biden State of the Union Address and the part I like best so far is his intention of raising Pell Grants.

Hello! Lately, I've been struggling with this assignment. (angle brackets represent closed interval) I figured out that: a) union = R intersection = {0} b) union = (0, 2) intersection = {1} I asked my prof about this and she explained to me that it should be shown that if a set is an...

in .y code: %union{ int ival; int *ivals; char id[20]; char *str; bool *b; } %start START %token COMMENT OP_PLUS OP_MINUS OP_DIV OP_MULT OP_OP OP_CP OP_DBLMULT OP_OC OP_CC OP_COMMA KW_AND KW_OR KW_NOT KW_EQUAL KW_LESS KW_NIL KW_LIST KW_APPEND KW_CONCAT KW_SET KW_DEFFUN...

Prove that if ##X## is a topological space, and ##S_i \subset X## is a finite collection of compact subspaces, then their union ##S_1 \cup \cdots \cup S_n## is also compact. ##S_i \subset X## is compact ##\therefore \forall S_i, \exists## a finite open cover ##\mathcal J_i=\{U_j\}_{j\in...

Hey! :giggle: We consider the set $X=\mathbb{R}\cup \{\star\}$, i.e. $X$ consists of $\mathbb{R}$ and an additional point $\star$. We say that $U\subset X$ is open if: (a) For each point $x\in U\cap \mathbb{R}$ there exists an $\epsilon>0$ such that $(x-\epsilon, x+\epsilon)\subset U$...

Hey! :giggle: The set of $2$-dimensional orthogonal matrices is given by $$O(2, \mathbb{R})=\{a\in \mathbb{R}^{2\times 2}\mid a^ta=u_2\}$$ Show the following: (a) $O(2, \mathbb{R})=D\cup S$ and $D\cap S=\emptyset$. It holds that $D=\{d_{\alpha}\mid \alpha\in \mathbb{R}\}$ and...

"Prove Theorem 7.1 about the probability of a union, using the 12.3 proof (see section 12.2) that involves indicator variables. Do not write the proof in full generality, only for three events. You should not use the product notation; you should write out all factors of the product." I'm taking...

Question 1: a) T' is the complementary event of T Therefore, T'=1-T In set T = {3,6,9,12} P(T)=4/12 =1/3 P(T')=1-1/3=2/3 b) The addition rule states; P(A ∪ B)=P(A)+P(B)-P(A⋂B) Therefore, P(S ∪ E) = P(S)+P(E)-P(S⋂E) S={1,4,9} P(S)=3/12=1/4 E={2,4,6,8,10,12} P(E)=6/12=1/2 (S⋂E)={4} P(S⋂E)=1/12...

Homework Statement:: x Relevant Equations:: x I stumbled upon the following example in the book - " How to prove it, A structured approach " ( 2nd edition) , Vellerman. Homework Statement:: He then asks to describe the set: ## \bigcup_{s \in S} L_{s} \, \backslash \, \bigcup_{s \in S}...

Let x ∈ A1 ∪ A2 then x ∈ A1 or x ∈ A2 If x ∈ A1, as A1 is open, there exists an r > 0 such that B(x,r) ⊂ A1⊂ A1 ∪ A2 and thus B(x,r) is an open set. Therefore A1 ∪ A2 is an open set. How does this prove that A1 ∪ A2 is an open set. It just proved that A1 ∪ A2 contains an open set; not that...

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For all ##n\in\mathbb{N}## we have ##\emptyset \in A_n##. Hence, ##\emptyset \in \mathcal{A}_\infty##. Let ##A \in \mathcal{A}_\infty##. Then ##A \in A_k## for some ##k\in\mathbb{N}##. So ##A^c \in A_k##. Hence, ##A^c \in \mathcal{A}_\infty##. Thus, ##\mathcal{A}_\infty## is closed under...

Is there a "ground union" type of fitting that can be installed in PEX lines without requiring additional adapters that screw into the fitting? I need to plumb a whole house sediment filter. Ideally, I like a union fitting that was PEX on one end and threaded on the other. This would allow...

My question concerns the portion of the proof stating, “...we set up a correspondence between the elements of U(A_n), for n in N, and a subset of S by making the element a correspond to (m, n) if A_m is the first set in which a appears, and a is the nth element of A_m.” In particular, I am...

Homework Statement Definition: A union of events A,B,C, . . . is an event consisting of all the outcomes in all these events. It occurs if any of A,B,C, . . . occurs, and therefore, corresponds to the word “or”: A or B or C or ... (Figure 2.1a). Homework EquationsThe Attempt at a Solution I'm...

Homework Statement Show that it is not necessarily true that the infinite union of closed sets is closed Homework EquationsThe Attempt at a Solution From intuition, I came up with the following counter-example: ##\displaystyle \bigcup_{n=2}^{\infty} \left[ \frac{1}{n}, \frac{n}{n+1} \right] =...

Homework Statement This is the exact phrasing form Linear Algebra Done Right by Axler: Prove that the union of three subspaces of V is a subspace of V if and only if one of the subspaces contains the other two. [This exercise is surprisingly harder than the previous exercise, possibly because...

Homework Statement Show that ##\displaystyle \bigcup_{n=2}^\infty \left[ \frac{1}{n} , \frac{n}{n+1} \right] = (0,1)##. Homework EquationsThe Attempt at a Solution I'm not sure how to show this rigorously. It is sufficient to note that ##\lim_{n\to\infty} \frac{1}{n} = 0## and that...

Here is a problem statement: Let ##H## be a proper subgroup of a finite group ##G##. Prove that the union of the conjugates of ##H## is not all of ##G##. I have proven this statement by considering the action of ##G## on ##\mathcal{P}(G)##. But this leads me to wonder: In the problem statement...

Homework Statement Let ##H_1 \le H_2 \le \cdots## be an ascending chain of subgroups of ##G##. Prove that ##H = \bigcup\limits_{i=1}^{\infty} H_{i}## is a subgroup of ##G##. Homework EquationsThe Attempt at a Solution Certainly ##H## is nonempty, since each subgroup ##H_i## has at least the...

I have this problem with understanding this specific notation could someone explain this notation & figure out how to solve this problem!? Suppose P(Q)=5/31 , P(R)= 7/31 , P(Q intersect R)=3/31 Find the value of: A) P(Q') B) P(Q union R) C) P(R')

Homework Statement From Linear Algebra and Its Applications, 5th Edition, David Lay Chapter 4, Section 1, Question 32 Let H and K be subspaces of a vector space V. The intersection of H and K is the set of v in V that belong to both H and K. Show that H ∩ K is a subspace of V. (See figure.)...

I'm trying to write an algorithm that will take in, as parameters, two rectangles R1 and R2 and calculate their union. R1 and R2 may be in rotated (independently), one may be completely inside the other, or they may not be overlapping at all. Example(Image): The algorithm I wrote currently...

I am going over some of my notes, and I cannot understand unions, here is the selection I am having trouble with How does the union of four different sets equal just one of the sets? Should the union of 4 sets be the four different sets instead of one. I am missing something fundamental to...

I am reading "Introduction to Set Theory" (Third Edition, Revised and Expanded) by Karel Hrbacek and Thomas Jech (H&J) ... ... I am currently focused on Chapter 1: Sets and, in particular on Section 3: The Axioms where Hrbacek and Jech set up an axiomatic systems (which they do NOT call ZFC ...

Homework Statement Let ##R## be a commutative ring with identity and suppose that ##A## is an ideal in ##R## contained in the finite union of prime ideals ##P_1 \cup \cdots \cup P_n##. Show that ##A \subseteq P_i## for some ##i##. Homework EquationsThe Attempt at a Solution The base case...

The complete graph K_n can be expressed as the union of k bipartite graphs iff n≤2^k I would simply like to know how to get 2^k.

there's a proof provided, but i want to know the intuition as to why it is 2^k.

Homework Statement Prove that a finite group is the union of proper subgroups if and only if the group is not cyclic. Homework Equations None The Attempt at a Solution [/B] " => " If the group, call it G, is a union of proper subgroups, then, for every subgroup, there is at least one...

Homework Statement Let ##A_n = (n − 1, n + 1)##, for all natural numbers n. Find, with proof, ##∪_{n≥1}A_n## Homework Equations What does that last statement mean? Union for n greater than or equal to one times the interval? The Attempt at a Solution I can't understand the question.

Given an indexed collection of sets A_x the disjoint union of these sets can be thought of as the ordinary union of the sets \{ x \} \times A_x for all x. That is, it is the set of all pairs (x, a) where a \in A_x. (Correct me at this point if my understanding of disjoint union is wrong.)...

I am a little confused here: a) The number 2 which is at the beginng of one set is closed. The number 2 is open at the beginning of the other set. b) The number 2 is closed of the beginning of a set which goes to infinity. The other set begins at 0 and goes to infinity (2 is an...

Re: Union and Intersection of Sets Hi, Please I need a help regarding Union of sets can anybody solve this A={1,2,3} and B={{1,2},3} then what is A Union B and A Intersect B Thanks

Prove that a-(b∩ c)=(a-b)u(a-c)

Homework Statement Show that if ##f: A \rightarrow B## and ##E,F \subseteq A##, then ##f(E \cup F) = f(E) \cup f(F)##, and ##f(E \cap F) \subseteq f(E) \cap f(F)##. Homework Equations ##f(E) := \{f(x)~|~ x \in E \}##. The Attempt at a Solution Okay, showing ##f(E \cup F) \subseteq f(E) \cup...

Homework Statement For each ##n \in \mathbb{N}##, let ##A_{n}=\left\{n\right\}##. What are ##\bigcup_{n\in\mathbb{N}}A_{n}## and ##\bigcap_{n\in\mathbb{N}}A_{n}##. Homework Equations The Attempt at a Solution I know that this involves natural numbers some how, I am just confused on a...

So I've been learning Set Theory by myself through Jech and Hrabeck textbook, and I'm having trouble understanding some axioms. 1. Homework Statement What exactly is the difference between the axiom of pair and axiom of union? From what I understood, the axiom of pair tells us that there is a...

I don't see how this is the case. Let ao and bo be members of [A,B] with ao<bo. Let {ai} be a strictly decreasing sequence, with each ai>A and {bi} be a strictly increasing sequencing with each bi<B. Let the limits of the two sequences be A and B, respectively. Then define Ii = [ai,bi]. It seems...

What's the meaning of \displaystyle \bigcup_{n=1}^{\infty} [5^{-n}, n]?

Homework Statement A and B are both spans. A = span(a1,a2,a3) B = span(b1,b2) Then how many elements would AUB have? Homework Equations N/A The Attempt at a Solution [/B] I'm almost certain the answer is 2, with the two elements being span(a1,a2,a3) and span(b1,b2) but I'm unsure as to...

Use inclusion-exclusion to find the number of ways to arrange the six numbers 1, 2, 3, 4, 5, 6 such that either 1 is immediately followed by 2, or 3 is immediately followed by 4, or 5 is immediately followed by 6. I believe that this can be solved using unions. By setting the sets to be the...

Homework Statement Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable. (Rudin: Principles of Mathematical Analysis, 2nd ed.) Homework Equations Every separable metric space has a countable base. The...

Homework Statement Prove that[/B] P(\cup_{i=1}^n E_i) \geq \max_i P(E_i) (1) for n≥1 Homework Equations I know that P(\cup_{i=1}^n E_i) \leq \sum_{i=1}^n P(E_i). The Attempt at a Solution I know when n=1, trivially P(E_1) \geq \max_1 P(E_1) =P(E_1). So I was hoping I could use induction to...

Homework Statement I would like to define pressure loses through transmission line to the transducer. In transmission line I have a reducing union, filter and L pipe. How I can find this differences? In case of frequency my reducing union reduce pipe diameter 6 mm to 3 mm how i can determine...

Hello. We all know that DeMorgan's Law is as follows: (A∪B)' = A'∩B' and (A∩B)' = A'∪B' where ' refers to the complement of a set and A and B are both sets. We also know that this can be extended to more than two terms. My question is whether or not the following is true: (A∩B∪C)' = A'∪B'∩C'...

I am working on a problem like this: Suppose ##\mathscr A_1 \subset \mathscr A_2 \subset \ldots## are sigma-algebras consisting of subsets of a set ##X##. Give an example that ##\bigcup_{i=1}^{\infty} \mathscr A_i## is not sigma-algebra. I was told to work along finite sigma-algebras on...

Hello! (Wave) I am looking at the proposition: If $(A_n)_{n \in \omega}$ is a sequence of sets and $(f_n)_{n \in \omega}$ is a sequence of functions then: for all $n \in \omega, f_n: \omega \overset{\text{ surjective }}{\rightarrow} A_n$ then there is a function $f: \omega \overset{\text{...

Hello! (Wave) When we have: $f(x)=0, \forall x \in A \wedge f(x)=0, \forall x \in B$, do we conclude that $f(x)=0, \forall x \in A \cap B$ or $f(x)=0, \forall x \in A \cup B$? (Thinking)