Question

Let S and T be nonempty subsets of R with the following property: s ≤ t...

Let S and T be nonempty subsets of R with the following property: s ≤ t for all s ∈ S and t ∈ T.

  1. (a) Show that S is bounded above and T is bounded below.

  2. (b) Prove supS ≤ inf T .

  3. (c) Given an example of such sets S and T where S ∩ T is nonempty.

  4. (d) Give an example of sets S and T where supS = infT and S ∩T is the empty set. In your explanation, make sure you justify why sup S = inf T .

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
let A be a nonempty subset of R that is bounded below. Prove that inf A...
let A be a nonempty subset of R that is bounded below. Prove that inf A = -sup{-a: a in A}
Suppose A ⊆ R is nonempty and bounded above and β ∈ R. Let A +...
Suppose A ⊆ R is nonempty and bounded above and β ∈ R. Let A + β = {a + β : a ∈ A} Prove that A + β has a supremum and sup(A + β) = sup(A) + β.
Is there a set A ⊆ R with the following property? In each case give an...
Is there a set A ⊆ R with the following property? In each case give an example, or a rigorous proof that it does not exist. d) Every real number is both a lower and an upper bound for A. (e) A is non-empty and 2inf(A) < a < 1 sup(A) for every a ∈ A.2 (f) A is non-empty and (inf(A),sup(A)) ⊆ [a+ 1,b− 1] for some a,b ∈ A and n > 1000.
Let f : [a,b] → R be a bounded function and let:             M = sup...
Let f : [a,b] → R be a bounded function and let:             M = sup f(x)             m = inf f(x)             M* =sup |f(x)|             m* =inf |f(x)| assuming you are taking values of x that lie in [a,b]. Is it true that M* - m* ≤ M - m ? If it is true, prove it. If it is false, find a counter example.
Prove: Let S be a bounded set of real numbers and let a > 0. Define...
Prove: Let S be a bounded set of real numbers and let a > 0. Define aS = {as : s ∈ S}. Show that inf(aS) = a*inf(S).
Discrete mathematics function relation problem Let P ∗ (N) be the set of all nonempty subsets...
Discrete mathematics function relation problem Let P ∗ (N) be the set of all nonempty subsets of N. Define m : P ∗ (N) → N by m(A) = the smallest member of A. So for example, m {3, 5, 10} = 3 and m {n | n is prime } = 2. (a) Prove that m is not one-to-one. (b) Prove that m is onto.
Let S be a nonempty set in Rn, and its support function be σS = sup{...
Let S be a nonempty set in Rn, and its support function be σS = sup{ <x,z> : z ∈ S}. let conv(S) denote the convex hull of S. Show that σS (x)= σconv(S) (x), for all x ∈ Rn
Real Analysis I Prove the following exercises (show all your work)- Exercise 1.1.1: Prove part (iii)...
Real Analysis I Prove the following exercises (show all your work)- Exercise 1.1.1: Prove part (iii) of Proposition 1.1.8. That is, let F be an ordered field and x, y,z ∈ F. Prove If x < 0 and y < z, then xy > xz. Let F be an ordered field and x, y,z,w ∈ F. Then: If x < 0 and y < z, then xy > xz. Exercise 1.1.5: Let S be an ordered set. Let A ⊂...
Let f : X → Y and suppose that {Ai}i∈I is an indexed collection of subsets...
Let f : X → Y and suppose that {Ai}i∈I is an indexed collection of subsets of X. Show that f[∩i∈IAi ] ⊆ ∩i∈I f[Ai ]. Give an example, using two sets A1 and A2, to show that it’s possible for the LHS to be empty while the RHS is non-empty.
(2) Let X be a set and < a linear order on X. Let S be...
(2) Let X be a set and < a linear order on X. Let S be a subset of X. Show that if S has a least element, then S has a unique least element. (3) Give an example, where S has no least element. (Be sure to specify what X, < and S are!) (4) Let X be a set and < a linear order on X. Let S be a subset of X which is bounded below. Show...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT