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Please do exercise 129:
Exercise 128: Define r:N + N by r(n) = next(next(n)). Let f:N...
4. Define a function f:N → Z by tof n/2 if n is even 1-(n +...
4. Define a function f:N → Z by tof n/2 if n is even 1-(n + 1)/2 if n is odd. f(n) = Show that f is a bijection. 11 ] 7. Let X = R XR and let R be a relation on X defined as follows ((x,y),(w,z)) ER 4 IC ER\ {0} (w = cx and z = cy.) Is R reflexive? Symmetric? Transitive? An equivalence relation? Explain each of your answers. Describe the equivalence classes [(0,0)]R and...
5. Let A = P(R). Define f : R → A by the formula f(x) =...
5. Let A = P(R). Define f : R → A by the formula f(x) = {y E RIy2 < x). (a) Find f(2). (b) Is f injective, surjective, both (bijective), or neither? Z given by f(u)n+l, ifn is even n - 3, if n is odd 6. Consider the function f : Z → Z given by f(n) = (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer
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....
Suppose f:N → N satisfies the recurrence f(n+1) = f(n) 7. Note that this is not...
Suppose f:N → N satisfies the recurrence f(n+1) = f(n) 7. Note that this is not enough information to define the function, since we don't have an initial condition. For each of the initial conditions below, find the value of f(7). a. f(0) = 1. $(7) = b. f(0) = 5. f(7) = c. f(0) = 19. f(7) = d. f(0) = 249. f(7) =
Theorem 3.14 Let (neN aand EneN be sequences in R. Let be in R# and suppose that x" → x, y, → oo,...
Finish the proof of Theorem 3.14.
Theorem 3.14 Let (neN aand EneN be sequences in R. Let be in R# and suppose that x" → x, y, → oo, and z" →-oo. . If -oo <x o, then +yn 2. If-oo x < 00, then x" + Zn →-00 4. If-oo x < 0, then xoY" →-00 and xnZn → oo. 5. If x is in R. then-→0and-" →0 Proof Note that the conditions in the different parts of the...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 ...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
Please Answer 135 Below Completely: Definition Let E-R and f : E-+ R be a function....
Please Answer 135 Below Completely:
Definition Let E-R and f : E-+ R be a function. For some p E E' we say that f is continuous at p if for any ε > 0, there exists a δ > 0 (which depends on ε) such that for any x E E with |x-Pl < δ we have If(x) -f(p)le KE. This is often called the rigorous δ-ε definition of continuity. A couple of things to note about this definition....
8 marks] Number representations. We define the predicate BT(n,): "r has a balanced ternary repres...
8 marks] Number representations. We define the predicate BT(n,): "r has a balanced ternary represen tation that contains n digits," where n eZ+ and z e Z. Equivalently BT(n,z) : 34, di, ,dn-le(-1,0,1), Σdi.gi=z Prove the following statement using induction on n 3a-1
8 marks] Number representations. We define the predicate BT(n,): "r has a balanced ternary represen tation that contains n digits," where n eZ+ and z e Z. Equivalently BT(n,z) : 34, di, ,dn-le(-1,0,1), Σdi.gi=z Prove the following...
Define a relation < on Z by m <n iff |m| < |n| or (\m| =...
Define a relation < on Z by m <n iff |m| < |n| or (\m| = |n| 1 m <n) (a) Prove that < is a partial order on Z. (b) A partial order R on a set S is called a total order (or linear order) iff (Vx, Y ES)(x + y + ((x, y) E R V (y,x) E R)) Prove that is a total order on Z. (c) List the following elements in <-increasing order. –5, 2,...
Number Theory 13 and 14 please! 13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is uni...
Number Theory
13 and 14 please!
13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
4. Define a function f:N → Z by tof n/2 if n is even 1-(n + 1)/2 if n is odd. f(n) = Show that f is a bijection. 11 ] 7. Let X = R XR and let R be a relation on X defined as follows ((x,y),(w,z)) ER 4 IC ER\ {0} (w = cx and z = cy.) Is R reflexive? Symmetric? Transitive? An equivalence relation? Explain each of your answers. Describe the equivalence classes [(0,0)]R and...
5. Let A = P(R). Define f : R → A by the formula f(x) = {y E RIy2 < x). (a) Find f(2). (b) Is f injective, surjective, both (bijective), or neither? Z given by f(u)n+l, ifn is even n - 3, if n is odd 6. Consider the function f : Z → Z given by f(n) = (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer
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....
Suppose f:N → N satisfies the recurrence f(n+1) = f(n) 7. Note that this is not enough information to define the function, since we don't have an initial condition. For each of the initial conditions below, find the value of f(7). a. f(0) = 1. $(7) = b. f(0) = 5. f(7) = c. f(0) = 19. f(7) = d. f(0) = 249. f(7) =
Finish the proof of Theorem 3.14.
Theorem 3.14 Let (neN aand EneN be sequences in R. Let be in R# and suppose that x" → x, y, → oo, and z" →-oo. . If -oo <x o, then +yn 2. If-oo x < 00, then x" + Zn →-00 4. If-oo x < 0, then xoY" →-00 and xnZn → oo. 5. If x is in R. then-→0and-" →0 Proof Note that the conditions in the different parts of the...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
Please Answer 135 Below Completely:
Definition Let E-R and f : E-+ R be a function. For some p E E' we say that f is continuous at p if for any ε > 0, there exists a δ > 0 (which depends on ε) such that for any x E E with |x-Pl < δ we have If(x) -f(p)le KE. This is often called the rigorous δ-ε definition of continuity. A couple of things to note about this definition....
8 marks] Number representations. We define the predicate BT(n,): "r has a balanced ternary represen tation that contains n digits," where n eZ+ and z e Z. Equivalently BT(n,z) : 34, di, ,dn-le(-1,0,1), Σdi.gi=z Prove the following statement using induction on n 3a-1
8 marks] Number representations. We define the predicate BT(n,): "r has a balanced ternary represen tation that contains n digits," where n eZ+ and z e Z. Equivalently BT(n,z) : 34, di, ,dn-le(-1,0,1), Σdi.gi=z Prove the following...
Define a relation < on Z by m <n iff |m| < |n| or (\m| = |n| 1 m <n) (a) Prove that < is a partial order on Z. (b) A partial order R on a set S is called a total order (or linear order) iff (Vx, Y ES)(x + y + ((x, y) E R V (y,x) E R)) Prove that is a total order on Z. (c) List the following elements in <-increasing order. –5, 2,...
Number Theory
13 and 14 please!
13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
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