Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be...
Recall that (a,b)⊆R means an open interval on the real number line:
(a,b)={x∈R|a<x<b}.
Let ≤ be the usual “less than or equal to” total order on the set
A=(−2,0)∪(0,2).
Consider the subset
B={−1/n | n∈N,n≥1}⊆A.
Determine an upper bound for B in A.. Then formally prove that B has no least upper bound in A by arguing that every element of A fails the criteria in the definition of least upper bound.
Note:
- make sure you are addressing the technical definitions of upper bound and least upper bound, not just your conceptions of these terms.
- Please note that the set A does not contain the number 0, and we are talking about upper bounds and the (potential for a) least upper bound for B in A. So your argument should not have anything to do with the number 0. Just pretend that number doesn't exist.
- For the least upper bound part, you are proving a "for every" statement. You need to provide a piece of evidence for each and every element in A that demonstrates that that element cannot be the least upper bound of B in A.
Homework Answers
Add Answer to:
Recall that (a,b)⊆R means an open interval on the real number
line:
(a,b)={x∈R|a<x<b}.
Let ≤ be...
1. Let A be a nonempty subset of R such that every number in A is...
1. Let A be a nonempty subset of R such that every number in A
is greater than 2 (NOTE: This doesn’t necessarily mean that A =
(2,∞)).
(a) Explain why A must have an infimum.
(b) Let c = inf(A). Prove that a∈A INTERSECTION (−∞,a] =
(−∞,c].
CAN SOMEONE PLZ HELP ME WITH THIS QUESTION.
1. Let A be a nonempty subset of R such that every number in A is greater than 2 (NOTE: This doesn't necessarily mean...
5- Recall that a set KCR is said to be compact if every open cover for...
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Let r be any rational number and define L = { x in Q: x <...
Let r be any rational number and define L = { x in Q: x < r }, the set of rational numbers less than r. Show that L is a Dedekind cut by proving the following properties: A. There exists a rational number x in L and there exists a rational number y not in L. ( This proves L is nonempty and L is not equal to Q) B. If x in L, then there exists z in...
help me. 5. consider set F(R):ff: f:R-R), but set all function with set real number in...
help me.
5. consider set F(R):ff: f:R-R), but set all function with set real number in domain and codomain. Show "addition" in any two function it.eCE(R) to produce new function such as given: ttgR2R which is every xER such as given:(tg)lx)-fx)+g(x), and any real number k ER, multiply it with any element f EF(R) to produce new function as given: kfRR in every value xER such as given:(k:0(x):-kfx)(observe it with multiply dua real number) (a) Show. FIR) ith addition and...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we ...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...
Let X be a continuous random variable defined on R. Then for any real number x...
Let X be a continuous random variable defined on R. Then for any real number x True ● False The staff at a small company includes: 2 secretaries, 12 technicians, 4 engineers, 2 executives, and 64 factory workers If a person is selected at random, what is the probability that he or she is a factory worker? 21 16 21 19 21 7 A 7 digit code number is generated by randomly selecting digits, with replacement, from the set(1.23 the...
Some Extra Definitions Recall that, for a nonrandom real number c, and a random variable X,...
Some Extra Definitions Recall that, for a nonrandom real number c, and a random variable X, we have Var (cX) = e Var (X). In this problem we'll generalize this property to linear combinations! Let be a vector of real nonrandom numbers, and let be a vector of random variables (sometimes called a random vector). Last, define the covariance matrix to be the matrix with all the covariances ar- ranged into a matrix. When we talk about taking the taking...
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the set S defined as follows: Va,bE S, arb if and only if every prime number that divides a is a factor of b and a S b. The r...
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the set S defined as follows: Va,bE S, arb if and only if every prime number that divides a is a factor of b and a S b. The relation T is a partial order relation (you do not need to prove this). Draw the Hasse diagram for T
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C++ OPTION A (Basic): Complex Numbers A complex number, c, is an ordered pair of real...
C++
OPTION A (Basic): Complex Numbers
A complex number, c,
is an ordered pair of real numbers
(doubles). For example, for any two real numbers,
s and t, we can form the complex number:
This is only part of what makes a complex number complex.
Another important aspect is the definition of special rules for
adding, multiplying, dividing, etc. these ordered pairs. Complex
numbers are more than simply x-y coordinates because of these
operations. Examples of complex numbers in this...
Please do exercise 129: Exercise 128: Define r:N + N by r(n) = next(next(n)). Let f:N...
Please do exercise 129:
Exercise 128: Define r:N + N by r(n) = next(next(n)). Let f:N → N be the unique function that satisfies f(0) = 2 and f(next(n)) =r(f(n)) for all n E N. 102 1. Prove that f(3) = 8. 2. Prove that 2 <f(n) for all n E N. Exercise 129: Define r and f as in Exercise 128. Assume that x + y. Define r' = {(x,y),(y,x)}. Let g:N + {x,y} be the unique function that...
1. Let A be a nonempty subset of R such that every number in A
is greater than 2 (NOTE: This doesn’t necessarily mean that A =
(2,∞)).
(a) Explain why A must have an infimum.
(b) Let c = inf(A). Prove that a∈A INTERSECTION (−∞,a] =
(−∞,c].
CAN SOMEONE PLZ HELP ME WITH THIS QUESTION.
1. Let A be a nonempty subset of R such that every number in A is greater than 2 (NOTE: This doesn't necessarily mean...
5- Recall that a set KCR is said to be compact if every open cover for K has a finite subcover 5-1) Use the above definition to prove that if A and B are two compact subsets of R then AUB is compact induction to show that a finite union of compact subsets of R is compact. 5-2) Now use 5-3) Let A be a nonempty finite subset of R. Prove that A is compact 5-4) Give an example of...
help me.
5. consider set F(R):ff: f:R-R), but set all function with set real number in domain and codomain. Show "addition" in any two function it.eCE(R) to produce new function such as given: ttgR2R which is every xER such as given:(tg)lx)-fx)+g(x), and any real number k ER, multiply it with any element f EF(R) to produce new function as given: kfRR in every value xER such as given:(k:0(x):-kfx)(observe it with multiply dua real number) (a) Show. FIR) ith addition and...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...
Let X be a continuous random variable defined on R. Then for any real number x True ● False The staff at a small company includes: 2 secretaries, 12 technicians, 4 engineers, 2 executives, and 64 factory workers If a person is selected at random, what is the probability that he or she is a factory worker? 21 16 21 19 21 7 A 7 digit code number is generated by randomly selecting digits, with replacement, from the set(1.23 the...
Some Extra Definitions Recall that, for a nonrandom real number c, and a random variable X, we have Var (cX) = e Var (X). In this problem we'll generalize this property to linear combinations! Let be a vector of real nonrandom numbers, and let be a vector of random variables (sometimes called a random vector). Last, define the covariance matrix to be the matrix with all the covariances ar- ranged into a matrix. When we talk about taking the taking...
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the set S defined as follows: Va,bE S, arb if and only if every prime number that divides a is a factor of b and a S b. The relation T is a partial order relation (you do not need to prove this). Draw the Hasse diagram for T
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the...
C++
OPTION A (Basic): Complex Numbers
A complex number, c,
is an ordered pair of real numbers
(doubles). For example, for any two real numbers,
s and t, we can form the complex number:
This is only part of what makes a complex number complex.
Another important aspect is the definition of special rules for
adding, multiplying, dividing, etc. these ordered pairs. Complex
numbers are more than simply x-y coordinates because of these
operations. Examples of complex numbers in this...
Please do exercise 129:
Exercise 128: Define r:N + N by r(n) = next(next(n)). Let f:N → N be the unique function that satisfies f(0) = 2 and f(next(n)) =r(f(n)) for all n E N. 102 1. Prove that f(3) = 8. 2. Prove that 2 <f(n) for all n E N. Exercise 129: Define r and f as in Exercise 128. Assume that x + y. Define r' = {(x,y),(y,x)}. Let g:N + {x,y} be the unique function that...
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