Q1 Let N be the set of all positive integers. Prove that each of the following collections of subsets of N is a topology. (i) Tị consists of N, Ø, and every set {1,2, ..., n}, for n any positive integer. (This is called the initial segment topology.) (ii) T2 consists of N, Ø, and every set {n, n+1,. }, for n any positive integer. ... (This is called the final segment topology.)

Q: Prove that the structure (even integers, +, *) is closed with respect to *.

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Q: 4- A subset [a, b] R is compact set in: a- (R, Td), b- (R, T), c-Both (R, Tu) and (R, T₁) d- No one.…

A: As per our guidelines we are supposed to answer only first part. Kindly repost other parts in the…

Q: Let Gbe an open set in R". Two points x, Y e G are said to be equivalent if they can be joined by a…

A: Since every open subset of Rn can be written of countable union of disjoint open balls and each ball…

Q: Let T = {0, En}, where En = {n +1, n + 2, n + 3, ...} with n E N, be a topology on N. Then the set…

A: Ans : False , because

Q: Let {E} be a countable collection of sets of real numbers. Then UEn ≤Σm* (En) n * m n

A: Definition of outer measure: The outer measure of a set E is defined as, m*(E)=inf∑j=1∞Qj, where…

Q: How do I write a set T such that S is an element of T and a proper subset of T?

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Q: Example 5. If A, B and C are three sets and U is the universal set such that n(U) = 1000, n(A) =…

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Q: 3.(1) List all the partitions of the set S = {1, 2, 3, 4}. (2) From (1), determine how many…

A: First five partitions are given in step one and the rest is given in step 2  1. 1, 2, 3, 4 2. 1, 2,…

Q: 3 about disjoint subsets A and B of a universal set U, except assume that n(U)=100, n(A)=30, and…

A: Given, n(U)=100 n(A)=30 n(B)=50 n(A′∪B)=?

Q: 1. (a) For a set A, P (A) is the set of subsets of A. Write down all the different elements of the…

A: (a)  The empty set, ϕ  is the subset of any set and a set is always a subset of itself. P(ϕ)={ϕ},…

Q: For each condition below, give an example of a set that satisfies that condition, or prove that one…

A: The given of the question is a list of conditions related to sets and subsets of real numbers, and…

Q: Determine if the following sets are closed under multiplication. a. (1,18) b. {2k + 9|keW} c.…

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Q: 1. a) Suppose A, B and C are sets. What do you need to show to prove the following using the…

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Q: A relation ''<'' is 1) İrreflexive. 2) Transitive. called a quasi-order on a set S if its: صواب ihi

A: We know that

Q: 5. For each pair of sets, give an explicit bijection from one to the other, showing that they have…

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Q: Invent an interesting set A with AC {a, b, c, d, e, ..., z} and |A|≥ 6 and another interesting set B…

A: As per our guideline, we are supposed to solve only first three subparts. Kindly repost other…

Q: Q2: Let the set under consideration be N for each n EN, define u, = {n, n+1, n+2, ...} and let 1 =…

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Q: Use the method of intersection of sets to compute the following. (a) GCD(12,16,8) and LCM(12,16,8)…

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Q: 5. Give an example of a collection (A,, of sets with the properties that: (a) A, # Ø for each i> 1,…

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Q: 1. (a) For a set A, P (A) is the set of subsets of A. Write down all the different elements of the…

A: Note: We are entitled to solve only the first question, unless specified 1) (a) We know the power…

Q: Prove T={ …,-56,-42,-28,-14,0,14,28,42,…} is a countable set

A: We have to prove the following -   T={ …,-56,-42,-28,-14,0,14,28,42,…} is a countable set.

Q: Let S be a set consisting of 10 elements. The number of tuples of the form (A, B) such that A and B…

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Q: Question 3. Consider the usual topology U on the set of real numbers. Describe the relative topology…

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Q: For an indexed collection of sets to have an intersection an element needs to be in every set right?…

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Q: Find the Car D={a, b, c, d, ..., z), n(D)=? U=universal set n(U')=? B={p people in the USA | p is a…

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Q: 1. Let A = {−30, -10, 10, 30, 50} and B = {-20, -10,0, 10} be subsets of the universal set ε = {10x:…

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Q: 3. Let F- (A.) be a finite collection of pairwise disjoint finite sets (n) Ume mathematioal…

A: Let F=Aii=1n be a finite collection of pairwise disjoint finite sets. (a) Prove ∪i=1nAi is finite…

Q: 15. We have seen that an infinite set can be put into a one-to-one correspondence with a proper…

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Q: Show that the set {2, 5, 8, 11, 14, . . . } is infinite.   Group of answer choices: Choose answer…

A: Topic-series

Q: I) We know that -5,-4,-3,-2,-1? and IN are well-ordered sets with respect to the trivial ordering.…

A: Solution: Given the sets -5,-4,-3,-2,-1 and ℕ are well-ordered with respect to the trivial ordering.…

Q: 3.) Nowe consider motrix cif possible) as a linear combination of the motrices in' ls. the set s…

A: Given that: S=3-333,3-245,6-3106To express the the following matrices in the linear combination of…

Q: A = {n EU | n is odd} and B = {n € U | 2n + 1 is a prime number}. Compute the size of the following…

A: Given Information: U=1,2,3,4,5,6,7,8,9 A=n∈U | n is odd B=n∈U | 2n+1 is a prime number

Q: 9. Given an infinite collection An, n = 1, 2, ... of intervals of the real line, their intersection…

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