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.

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Colby Messinger answered 2 years ago

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

Let A = {a1, a2, a3, . . . , an} be a nonempty set of...

Let A = {a1, a2, a3, . . . , an} be a nonempty set of n distinct natural numbers. Prove that there exists a nonempty subset of A for which the sum of its elements is divisible by n.

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...

1. Let α < β be real numbers and N ∈ N. (a). Show that if...

1. Let α < β be real numbers and N ∈ N. (a). Show that if β − α > N then there are at least N distinct integers strictly between β and α. (b). Show that if β > α are real numbers then there is a rational number q ∈ Q such β > q > α. *********************************************************************************************** 2. Let x, y, z be real numbers.The absolute value of x is defined by |x|= x, if x ≥...

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...

If a set K that is a subset of the real numbers is closed and bounded,...

If a set K that is a subset of the real numbers is closed and bounded, then it is compact.

PROOFS: 1. State the prove The Density Theorem for Rational Numbers 2. Prove that irrational numbers are dense in the set of real numbers

PROOFS: 1. State the prove The Density Theorem for Rational Numbers 2. Prove that irrational numbers are dense in the set of real numbers 3. Prove that rational numbers are countable 4. Prove that real numbers are uncountable 5. Prove that square root of 2 is irrational

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