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2. [14 marks] Rational Numbers The rational numbers, usually denoted Q are the set {n E R 3p, q ZAq&0An= Note that...
6 Set Operations • R, the set of real numbers • Q, the set of rational...
6 Set Operations • R, the set of real numbers • Q, the set of rational numbers: {a/b: ab € ZAb0} • Z, the set of integers: {..., -2,-1,0,1,2,...} • N, the set of natural numbers: {0,1,2,3,...} (e) What is NUQ? Q? (f) What kind of numbers are in R (g) If SCT, what is S T?
Suppose we tried to apply our real analysis definitions/methods to the set of rational numbers Q....
Suppose we tried to apply our real analysis definitions/methods
to the
set of rational numbers Q. In other words, in the definitions, we
only
consider rational numbers. E.g., [0, 1] now means [0, 1] ∩ Q, etc.
In
this setting:
(a) Find an open cover of [0, 1] that contains no finite subcover.
Hint:
Fix an irrational number α ∈ [0, 1] (as a subset of the reals
now!)
and for each (rational) q ∈ [0, 1] look for an...
Suppose we tried to apply our real analysis definitions/methods to the set of rational numbers Q....
Suppose we tried to apply our real analysis definitions/methods to the set of rational numbers Q. In other words, in the definitions, we only consider rational numbers. E.g., [0, 1] now means [0, 1] n Q, etc. In this setting: (a) Find an open cover of [0, 1] that contains no finite subcover. Hint: Fix an irrational number a € [0, 1] (as a subset of the reals now!) and for each (rational) qe [0, 1] look for an open...
B. Let p and q be distinct positive prime numbers. Set a p+ (a) Find a...
B. Let p and q be distinct positive prime numbers. Set a p+ (a) Find a monic polynomial f(x) EQlr of degree 4 such that f(a) 0. (b) Explain why part (a) shows that (Q(a):QS4 (c) Note: In order to be sure that IQ(α) : Q-4, we would need to know that f is irreducible. (Do not attempt it, though). Is it enough to show that f(x) has no rational roots? (d) Show V pg E Q(α). Does it follow...
It would be very useful to have a theory about computability of functions R" -> R. Given that there Q.2 are an u...
It would be very useful to have a theory about computability of functions R" -> R. Given that there Q.2 are an uncountable number of real numbers, we would need to start with a definition of which numbers are themselves computable. A natural definition would be that a real number x is computable if it is the limit of computable sequence of rational numbers (so that we can compute it to whatever accuracy we like). More carefully V a Definition:...
PROBLEM 2: THE INDICATOR FUNCTION OF THE RATIONAL NUMBERS For a while, it was believed that any given function should be mostly continuous. This is reasonable, given the types of functions one typica...
PROBLEM 2: THE INDICATOR FUNCTION OF THE RATIONAL NUMBERS For a while, it was believed that any given function should be mostly continuous. This is reasonable, given the types of functions one typically sees in Calculus courses, where the worst case scenario involves a function that is defined piecewise and is continuous everywhere, except for some finite set of discontinuities, where the value of the function drops or jumps. It was also believed that every function should be integrable, which...
real analysis 1,2,3,4,8please 5.1.5a Thus iff: I→R is differentiable on n E N. is differentiable on / with g'...
real analysis
1,2,3,4,8please
5.1.5a
Thus iff: I→R is differentiable on n E N. is differentiable on / with g'(e) ()ain tained from Theorem 5.1.5(b) using mathematical induction, TOu the interal 1i then by the cho 174 Chapter s Differentiation ■ EXERCISES 5.1 the definition to find the derivative of each of the following functions. I. Use r+ 1 2. "Prove that for all integers n, O if n is negative). 3. "a. Prove that (cosx)--sinx. -- b. Find the derivative...
Real analysis 10 11 12 13 please (r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and li...
Real analysis
10 11 12 13 please
(r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and limf(x)-L. Prove that lim)ILI. h If, in addition, )o for all x E E, prove that lim b. Prove that lim (f(x))"-L" for each n E N. ethe limit theorems, examples, and previous exercises to find each of the following limits. State which theo- rems, examples, or exercises are used in each case....
Problem 5. (20 pts) Let r,n N be two natural numbers with r < n. An...
Problem 5. (20 pts) Let r,n N be two natural numbers with r < n. An r x n matrix M consisting of r rows and n columns is said to be a Latin rectangle of size (r, n), if all the entries My belong to the set {1,2,3,..., n), for 1Si<T, 1Sj<T, and the same number does not appear twice in any row or in any column. By defini- tion, a Latin square is a Latin rectangle of size...
Problem 25 please -Sesin(2x)-9ecos(2x). 21. W = Span(B), where Br(x2e-4x , xe®, e-4x); f(x)--5x2r" + 2e-4-1e...
Problem 25 please
-Sesin(2x)-9ecos(2x). 21. W = Span(B), where Br(x2e-4x , xe®, e-4x); f(x)--5x2r" + 2e-4-1e 22. W= Span(B),where B= ({x25, x5*, 5x)); f(x)--4x2 5x+9s5x-2(5x). 3 W Span(B), where B (Exsin(2x), xcos(2x), sin(2x), cos(2x)y): f(x) = 4x sin(2x) + 9x cos(20-5 sin(2x) + 8 cos(2x). 24, In Exercise 21 of Section 3.6, we constructed the matrix [D, of the derivative operator D on W- Span(B), where B e sin(bx), e" cos(bx)): Dls a a. Find [D 1g and [D'lg: Observe...
6 Set Operations • R, the set of real numbers • Q, the set of rational numbers: {a/b: ab € ZAb0} • Z, the set of integers: {..., -2,-1,0,1,2,...} • N, the set of natural numbers: {0,1,2,3,...} (e) What is NUQ? Q? (f) What kind of numbers are in R (g) If SCT, what is S T?
Suppose we tried to apply our real analysis definitions/methods
to the
set of rational numbers Q. In other words, in the definitions, we
only
consider rational numbers. E.g., [0, 1] now means [0, 1] ∩ Q, etc.
In
this setting:
(a) Find an open cover of [0, 1] that contains no finite subcover.
Hint:
Fix an irrational number α ∈ [0, 1] (as a subset of the reals
now!)
and for each (rational) q ∈ [0, 1] look for an...
Suppose we tried to apply our real analysis definitions/methods to the set of rational numbers Q. In other words, in the definitions, we only consider rational numbers. E.g., [0, 1] now means [0, 1] n Q, etc. In this setting: (a) Find an open cover of [0, 1] that contains no finite subcover. Hint: Fix an irrational number a € [0, 1] (as a subset of the reals now!) and for each (rational) qe [0, 1] look for an open...
B. Let p and q be distinct positive prime numbers. Set a p+ (a) Find a monic polynomial f(x) EQlr of degree 4 such that f(a) 0. (b) Explain why part (a) shows that (Q(a):QS4 (c) Note: In order to be sure that IQ(α) : Q-4, we would need to know that f is irreducible. (Do not attempt it, though). Is it enough to show that f(x) has no rational roots? (d) Show V pg E Q(α). Does it follow...
It would be very useful to have a theory about computability of functions R" -> R. Given that there Q.2 are an uncountable number of real numbers, we would need to start with a definition of which numbers are themselves computable. A natural definition would be that a real number x is computable if it is the limit of computable sequence of rational numbers (so that we can compute it to whatever accuracy we like). More carefully V a Definition:...
PROBLEM 2: THE INDICATOR FUNCTION OF THE RATIONAL NUMBERS For a while, it was believed that any given function should be mostly continuous. This is reasonable, given the types of functions one typically sees in Calculus courses, where the worst case scenario involves a function that is defined piecewise and is continuous everywhere, except for some finite set of discontinuities, where the value of the function drops or jumps. It was also believed that every function should be integrable, which...
real analysis
1,2,3,4,8please
5.1.5a
Thus iff: I→R is differentiable on n E N. is differentiable on / with g'(e) ()ain tained from Theorem 5.1.5(b) using mathematical induction, TOu the interal 1i then by the cho 174 Chapter s Differentiation ■ EXERCISES 5.1 the definition to find the derivative of each of the following functions. I. Use r+ 1 2. "Prove that for all integers n, O if n is negative). 3. "a. Prove that (cosx)--sinx. -- b. Find the derivative...
Real analysis
10 11 12 13 please
(r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and limf(x)-L. Prove that lim)ILI. h If, in addition, )o for all x E E, prove that lim b. Prove that lim (f(x))"-L" for each n E N. ethe limit theorems, examples, and previous exercises to find each of the following limits. State which theo- rems, examples, or exercises are used in each case....
Problem 5. (20 pts) Let r,n N be two natural numbers with r < n. An r x n matrix M consisting of r rows and n columns is said to be a Latin rectangle of size (r, n), if all the entries My belong to the set {1,2,3,..., n), for 1Si<T, 1Sj<T, and the same number does not appear twice in any row or in any column. By defini- tion, a Latin square is a Latin rectangle of size...
Problem 25 please
-Sesin(2x)-9ecos(2x). 21. W = Span(B), where Br(x2e-4x , xe®, e-4x); f(x)--5x2r" + 2e-4-1e 22. W= Span(B),where B= ({x25, x5*, 5x)); f(x)--4x2 5x+9s5x-2(5x). 3 W Span(B), where B (Exsin(2x), xcos(2x), sin(2x), cos(2x)y): f(x) = 4x sin(2x) + 9x cos(20-5 sin(2x) + 8 cos(2x). 24, In Exercise 21 of Section 3.6, we constructed the matrix [D, of the derivative operator D on W- Span(B), where B e sin(bx), e" cos(bx)): Dls a a. Find [D 1g and [D'lg: Observe...
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