Question

1. Let A and B be subsets of R, each of which A and B be...

1. Let A and B be subsets of R, each of which A and B be subsets of R, each of which has a minimum element. Prove that if A ⊆ B, then min A ≥ min B.

2.. Let a and b be real numbers such that a < b. Prove that a < a + b / 2 < b. This number a + b / 2 is called the arithmetic mean of a and b.

3.. Let a, b and c be real numbers such that a < b and a < c. Prove that b < c if and only if [a, b) ⊂ [a, c)..

4. Let A ⊆ R, c ∈ R, and suppose A has a maximum element. Prove that c is an upper bound for A if and only if c ≥ max A.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let A ={1-1/n | n is a natural number} Prove that 0 is a lower bound...
Let A ={1-1/n | n is a natural number} Prove that 0 is a lower bound and 1 is an upper bound:  Start by taking x in A.  Then x = 1-1/n for some natural number n.  Starting from the fact that 0 < 1/n < 1 do some algebra and arithmetic to get to 0 < 1-1/n <1. Prove that lub(A) = 1:  Suppose that r is another upper bound.  Then wts that r<= 1.  Suppose not.  Then r<1.  So 1-r>0....
2. Let A, B, C be subsets of a universe U. Let R ⊆ A ×...
2. Let A, B, C be subsets of a universe U. Let R ⊆ A × A and S ⊆ A × A be binary relations on A. i. If R is transitive, then R−1 is transitive. ii. If R is reflexive or S is reflexive, then R ∪ S is reflexive. iii. If R is a function, then S ◦ R is a function. iv. If S ◦ R is a function, then R is a function
Let A, B be non-empty subsets of R. Define A + B = {a + b...
Let A, B be non-empty subsets of R. Define A + B = {a + b | a ∈ A and b ∈ B}. (a) If A = (−1, 2] and B = [1, 4], what is A + B?
Let X be a set and A a σ-algebra of subsets of X. (a) A function...
Let X be a set and A a σ-algebra of subsets of X. (a) A function f : X → R is measurable if the set {x ∈ X : f(x) > λ} belongs to A for every real number λ. Show that this holds if and only if the set {x ∈ X : f(x) ≥ λ} belongs to A for every λ ∈ R. (b) Let f : X → R be a function. (i) Show that if...
1. Let A ⊆ R and p ∈ R. We say that A is bounded away...
1. Let A ⊆ R and p ∈ R. We say that A is bounded away from p if there is some c ∈ R+ such that |x − p| ≥ c for all x ∈ A. Prove that A is bounded away from p if and only if p not equal to A and the set n { 1 / |x−p| : x ∈ A} is bounded. 2. (a) Let n ∈ natural number(N) , and suppose that k...
Let S be a finite set and let P(S) denote the set of all subsets of...
Let S be a finite set and let P(S) denote the set of all subsets of S. Define a relation on P(S) by declaring that two subsets A and B are related if A and B have the same number of elements. (a) Prove that this is an equivalence relation. b) Determine the equivalence classes. c) Determine the number of elements in each equivalence class.
A. Let p and r be real numbers, with p < r. Using the axioms of...
A. Let p and r be real numbers, with p < r. Using the axioms of the real number system, prove there exists a real number q so that p < q < r. B. Let f: R→R be a polynomial function of even degree and let A={f(x)|x ∈R} be the range of f. Define f such that it has at least two terms. 1. Using the properties and definitions of the real number system, and in particular the definition...
a) Let f : [a, b] −→ R and g : [a, b] −→ R be...
a) Let f : [a, b] −→ R and g : [a, b] −→ R be differentiable. Then f and g differ by a constant if and only if f ' (x) = g ' (x) for all x ∈ [a, b]. b) For c > 0, prove that the following equation does not have two solutions. x3− 3x + c = 0, 0 < x < 1 c) Let f : [a, b] → R be a differentiable function...
1. (a) Let S be a nonempty set of real numbers that is bounded above. Prove...
1. (a) Let S be a nonempty set of real numbers that is bounded above. Prove that if u and v are both least upper bounds of S, then u = v. (b) Let a > 0 be a real number. Define S := {1 − a n : n ∈ N}. Prove that if epsilon > 0, then there is an element x ∈ S such that x > 1−epsilon.
Let R*= R\ {0} be the set of nonzero real numbers. Let G= {2x2 matrix: row...
Let R*= R\ {0} be the set of nonzero real numbers. Let G= {2x2 matrix: row 1(a b) row 2 (0 a) | a in R*, b in R} (a) Prove that G is a subgroup of GL(2,R) (b) Prove that G is Abelian
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT
Active Questions
  • Which of the following is true about the characteristics of abstract data types? i) It creates...
    asked 9 minutes ago
  • 1. For each pseudo-code below derive the simplified asymptotic running time in Q(?) notation. (1) for...
    asked 13 minutes ago
  • A manufacturer of industrial solvent guarantees its customers that each drum of solvent they ship out...
    asked 22 minutes ago
  • Which of the following data structure can’t store non-homogeneous data elements? A) Arrays B) Linked Lists...
    asked 32 minutes ago
  • Provide a recursive definition of some sequence of numbers or function (e.g. log, exponent, polynomial). Choose...
    asked 33 minutes ago
  • 4. Prove explicitly that congruence modulo 4 is an equivalence relation. Then list the equivalence classes....
    asked 45 minutes ago
  • a.) The photoelectric effect is the basis of the spectroscopic technique known as photoelectron spectroscopy. An...
    asked 46 minutes ago
  • [The following information applies to the questions displayed below.] Washington Warehouse is a small retail business...
    asked 53 minutes ago
  • Given the following two sets of quotations by two currency dealers: Dealer A                               &n
    asked 55 minutes ago
  • The programming language for this exercise must be on C++. The exercise is the following: Design...
    asked 55 minutes ago
  • Continuity, differentiability questions: • Give an example of a function f: R → R that is...
    asked 55 minutes ago
  • What are some non-verbal differences in cultures, different countries, etc that can be mis-interpreted in a...
    asked 1 hour ago