代写MAT 137Y: Calculus with proofs Assignment 1调试数据库编程
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Assignment 1
1. Let S be a circle of radius 5. Let d(a, b) denote the distance between two points a and b.
Note: (0, 10) is the open interval of real numbers between 0 and 10. Mathematically, a point a is on the circle means the distance between this point and the center of the circle is equal to the radius. A point a is inside the circle means the distance between this point and the center of the circle is strictly less than the radius. Visually, here is a picture to show you one point a on a circle vs one point a inside a circle.
Below are five claims. Which ones are true and which ones are false? If a claim is true, prove it. If a claim is false, provide a counterexample and a justification of how the counterexample shows the claim is false.
(a) For all a inside S there is a c ∈ (0, 10) such that d(a, b) = c for every b on S.
This statement is O True O False |
(b) There exists a b on S such that for all c ∈ (0, 10) there is an a inside S such that d(a, b) = c.
This statement is O True O False |
(c) For every a inside S there exists a b on S such that for all c ∈ (0, 10), d(a, b) c.
This statement is O True O False |
(d) There exists a c > 10 such that for any b on S there exists an a inside S such that d(a, b) = c.
This statement is o True o False |
(e) For all c ∈ (0, 5), there exists an a inside S such that d(a, b) > c for every b on S.
This statement is o True o False |
2. Write down the negations of 1(a) and 1(d).
(a) What is the negation of statement 1(a)?
(b) What is the negation of statement 1(d)?
3. In this problem, assume all functions have domain R. I define two new concepts.
(i) We say that function f is better than function g if
8x ∈ R, 9y ∈ R s.t. x < y AND f (y) - g(y) < 2/1 .
(ii) We say that function f is simpler than function g if
9x ∈ R s.t. 8y ∈ R, x < y =) f (y) - g(y) < 2/1 .
(a) Write down the following definitions.
We say that function f is NOT better than function g if
We say that function f is NOT simpler than function g if
Below are three claims.Which ones are true and which ones are false? If a claim is true, prove it. If a claim is false, provide a counterexample and a justification of how the counterexample shows the claim is false. Hint: you may start this question by drawing pictures.
(b) If f and g are any two non-constant functions and f is better than g, then f is simpler than g.
This statement is O True O False |
(c) If f and g are any two non-constant functions and f is simpler than g, then f is better than g.
This statement is O True O False |
(d) If f , g and h are any three non-constant functions and f is better than g and g is better than h, then f is better than h.
This statement is O True O False |
4. Let cbe a given positive integer with c ≥ 2. Let {an} be the sequence of Fibonacci numbers. Fibonacci numbers are a list of infinitely many integers such that a1 = 1, a2 = 1, a3 = 2, a4 = 3, · · · and an+1 = an + an-1 for any n > 1. For example, a5 = a4 + a3 = 2 + 3 = 5.
Prove that for every integer n ≥ 1,
ac+n = acan+1 + ac — 1 an. (1)
Note: c remains fixed but arbitrary throughout the proof. The induction is on n not c.