Let A be a set of real numbers. We say that A is an open set...

Let A be a set of real numbers. We say that A is an open set if for every x₀ ∈ A there is some δ > 0 (which might depend on x₀) such that (x₀ − δ, x₀ + δ) ⊆ A. Show that a set B of real numbers is closed if and only if B is the complement of some open set A

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Colby Messinger answered 1 year ago

Let A = {a1,...,an} be a set of real numbers such that ai >= 1 for...

Let A = {a1,...,an} be a set of real numbers such that ai >= 1 for all i, and let I be an open interval of length 1. Use Sperner’s Theorem to prove an upper bound for the number of subsets of A whose elements sum to a number inside the interval I.

Given an array A of n distinct real numbers, we say that a pair of numbers...

Given an array A of n distinct real numbers, we say that a pair of numbers i, j ∈ {0, . . . , n−1} form an inversion of A if i < j and A[i] > A[j]. Let inv(A) = {(i, j) | i < j and A[i] > A[j]}. Answer the following: (a) How small can the number of inversions be? Give an example of an array of length n with the smallest possible number of inversions. (b)...

A sequence is just an infinite list of numbers (say real numbers, we often denote these...

A sequence is just an infinite list of numbers (say real numbers, we often denote these by a0,a1,a2,a3,a4,.....,ak,..... so that ak denotes the k-th term in the sequence. It is not hard to see that the set of all sequences, which we will call S, is a vector space. a) Consider the subset, F, of all sequences, S, which satisfy: ∀k ≥ 2,a(sub)k = a(sub)k−1 + a(sub)k−2. Prove that F is a vector subspace of S. b) Prove that if...

Let V be the set of all ordered triples of real numbers. For u = (u1,...

Let V be the set of all ordered triples of real numbers. For u = (u1, u2, u3) and v = (v1, v2, v3), we define the following operations of addition and scalar multiplication on V : u + v = (u1 + v1, u2 + v2 − 1, u3 + v3 − 2) and ku = (ku1, ku2, ku3). For example, if u = (1, 0, 3), v = (2, 1, 1), and k = 2 then u +...

Let S be the set of all ordered pairs of real numbers. Deﬁne scalar multiplication and...

Let S be the set of all ordered pairs of real numbers. Deﬁne scalar multiplication and addition on S by: α(x1,x2)=(αx1,αx2) (x1,x2)⊕(y1,y2)=(x1 +y1,0) We use the symbol⊕to denote the addition operation for this system in order to avoid confusion with the usual addition x+y of row vectors. Show that S, together with the ordinary scalar multiplication and the addition operation⊕, is not a vector space. Test ALL of the eight axioms and report which axioms fail to hold.

Let V be the set of all ordered pairs of real numbers. Consider the following addition...

Let V be the set of all ordered pairs of real numbers. Consider the following addition and scalar multiplication operations V. Let u = (u1, u2) and v = (v1, v2). Show that V is not a vector space. • u ⊕ v = (u1 + v1 + 1, u2 + v2 + 1 ) • ku = (ku1 + k − 1, ku2 + k − 1) 1)Show that the zero vector is 0 = (−1, −1). 2)Find the...

Let S be a set of n numbers. Let X be the set of all subsets...

Let S be a set of n numbers. Let X be the set of all subsets of S of size k, and let Y be the set of all ordered k-tuples (s1, s2, , sk) such that s1 < s2 < < sk. That is, X = {{s1, s2, , sk} | si S and all si's are distinct}, and Y = {(s1, s2, , sk) | si S and s1 < s2 < < sk}. (a) Define a one-to-one correspondence f : X → Y. Explain...

Suppose we define a relation ~ on the set of nonzero real numbers R* = R\{0}...

Suppose we define a relation ~ on the set of nonzero real numbers R* = R\{0} by for all a , b E R*, a ~ b if and only if ab>0. Prove that ~ is an equivalence relation. Find the equivalence class [8]. How many distinct equivalence classes are there?

We say that an infinite sequence a0,a1,a2,a3,… of real numbers has the limit L if for...

We say that an infinite sequence a0,a1,a2,a3,… of real numbers has the limit L if for every strictly positive number ε, there is a natural number n such that all the elements an,an+1,an+2,… are within distance ε of the value L. In this case, we write lim a = L. Express the condition that lim a = L as a formula of predicate logic. Your formula may use typical mathematical functions like + and absolute value and mathematical relations like...

Let A and B be sets of real numbers such that A ⊂ B. Find a...

Let A and B be sets of real numbers such that A ⊂ B. Find a relation among inf A, inf B, sup A, and sup B. Let A and B be sets of real numbers and write C = A ∪ B. Find a relation among sup A, sup B, and sup C. Let A and B be sets of real numbers and write C = A ∩ B. Find a relation among sup A, sup B, and sup...

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