Question

Let S be the collection of all sequences of real numbers and define a relation on...

Let S be the collection of all sequences of real numbers and define a relation on S by {xn} ∼ {yn} if and only if {xn − yn} converges to 0.

a) Prove that ∼ is an equivalence relation on S.

b) What happens if ∼ is defined by {xn} ∼ {yn} if and only if {xn + yn} converges to 0?

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 R be a relation on set RxR of ordered pairs of real numbers such that...
Let R be a relation on set RxR of ordered pairs of real numbers such that (a,b)R(c,d) if a+d=b+c. Prove that R is an equivalence relation and find equivalence class [(0,b)]R
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).
Prove: Let x and y be bounded sequences such that xn ≤ yn for all n...
Prove: Let x and y be bounded sequences such that xn ≤ yn for all n ∈ N. Then lim supn→∞ xn ≤ lim supn→∞ yn and lim infn→∞ xn ≤ lim infn→∞ yn.
Let R be the relation on the set of real numbers such that xRy if and...
Let R be the relation on the set of real numbers such that xRy if and only if x and y are real numbers that differ by less than 1, that is, |x − y| < 1. Which of the following pair or pairs can be used as a counterexample to show this relation is not an equivalence relation? A) (1, 1) B) (1, 1.8), (1.8, 3) C) (1, 1), (3, 3) D) (1, 1), (1, 1.5)
Let p and q be any two distinct prime numbers and define the relation a R...
Let p and q be any two distinct prime numbers and define the relation a R b on integers a,b by: a R b iff b-a is divisible by both p and q. I need to prove that: a) R is an equivalence relation. (which I have) b) The equivalence classes of R correspond to the elements of  ℤpq. That is: [a] = [b] as equivalence classes of R if and only if [a] = [b] as elements of ℤpq I...
I have a discrete math question. let R be a relation on the set of all...
I have a discrete math question. let R be a relation on the set of all real numbers given by cry if and only if x-y = 2piK for some integer K. prove that R is an equivalence relation.
Let p and q be any two distinct prime numbers and define the relation a R...
Let p and q be any two distinct prime numbers and define the relation a R b on integers a,b by: a R b iff b-a is divisible by both p and q. For this relation R: Prove that R is an equivalence relation. you may use the following lemma: If p is prime and p|mn, then p|m or p|n
Consider the relation on the real numbers R. a ~ b if (a−b) ∈ Z. (Z...
Consider the relation on the real numbers R. a ~ b if (a−b) ∈ Z. (Z is the whole integers.) 1) Give two real numbers that are in the same equivalence class. 2) Give two real numbers that are not in the same equivalence class. 3) Prove that this relation is an equivalence relation.
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.
Define a relation ? over the set of nonzero rational numbers as ??? if and only...
Define a relation ? over the set of nonzero rational numbers as ??? if and only if ??>0. Is R an equivalence relation?
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT