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 2 years 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 +...

For any sequence (xn) of real numbers, we say that (xn) is increasing iff for all...

For any sequence (xn) of real numbers, we say that (xn) is increasing iff for all n, m ∈ N, if n < m, then xn < xm. Prove that any increasing sequence that is not Cauchy must be unbounded. (Here, “unbounded” just means that xn eventually gets larger than any given real number). Then, show that any increasing sequence that is bounded must converge

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...

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