It is given that every nonzero rational number can be represented uniquely by a triple , where a, b nonzero and gcd of a and b is 1. Now the required triples form a subset of . Finite products of with is still countable. Subsets of countable sets are countable, hence the set of triples representing all nonzero rational numbers is countable. Adding one more elements (0) still keeps the set countable. Hence the set of all rational numbers is countable.
unique representation in the form -1)*a Every where k E 0,1} and a,b e N with a,b/ 0 and the greatest common divisor of...
8.3.1 Prove the following: (a) Every infinite series of the form 0O 72 (8.2) using appropriate choices for the coefficients Ibn], with the restrictioin that either bn- 0 or bn 1 is true for every n, represents a number in the interval [0, 1), with the exception that if bn n then the sum of the series is exactly 1. (b) Every real number in the interval [0,1) can be represented using a binary expansion, that is, can be represented...
all parts A-E please. Problem 8.43. For sake of a contradiction, assume the interval (0,1) is countable. Then there exists a bijection f : N-> (0,1). For each n є N, its image under f is some number in (0, 1). Let f(n) :-0.aina2na3n , where ain 1s the first digit in the decimal form for the image of n, a2 is the second digit, and so on. If f (n) terminates after k digits, then our convention will be...
1. (10 points) GCD Algorithm The greatest common divisor of two integers a and b where a 2 b is equal to the greatest common divisor of b and (a mod b). Write a program that implements this algorithm to find the GCD of two integers. Assume that both integers are positive. Follow this algorithm: 1. Call the two integers large and small. 2. If small is equal to 0: stop: large is the GCD. 3. Else, divide large by...
where Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) < 00, then {x E X (z) > 0} is a countable set. (HINT: Show that for every k E N the set {x E X | f(x) > k-1} is finite.) f(x)-sup f(x) | F is any finite subset of X TEF Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) 0} is a countable set. (HINT: Show that...
4. Show that any integer n 2 1 has a representation in the form with 'digits" d in the range di e [0,1,.i. For example, 2019-11+21+4.4!+4.51+2.6! 4. Show that any integer n 2 1 has a representation in the form with 'digits" d in the range di e [0,1,.i. For example, 2019-11+21+4.4!+4.51+2.6!
Use R language to program Problem 1: Greatest Common Divisor (GCD) Please write two functions, g edi ) and gcdr , which both take two integers a, b and calculates their greatest common divisor (GCD) using the Euclidean algorithm gcdi () should do so using iteration while gcdr () should use recursion. Then write a third function, gcd(), which takes two integers a, band an optional third argument nethod which takes a charater string containing either "iterative" or "recursive", with...
Using Python!!! Write a recursive function gcd(m,n) that returns the greatest common divisor of a pair of numbers. The gcd of m and n is the largest number that divides both m and n. If one of the numbers is 0, then the gcd is the other number. If m is greater than or equal to n, then the gcd of m and n is the same as the gcd of n and m-n. If n is greater than m,...
Integral: If you know all about it you should be easy to prove..... Let f:[a,b]→R and g:[a,b]→R be two bounded functions. Suppose f≤g on [a,b]. Use the information to prove thatL(f)≤L(g)andU(f)≤U(g). Information: g : [0, 1] —> R be defined by if x=0, g(x)=1; if x=m/n (m and n are positive integer with no common factor), g(x)=1/n; if x doesn't belong to rational number, g(x)=0 g is discontinuous at every rational number in[0,1]. g is Riemann integrable on [0,1] based...
Suppose f : B(0.1) C is holomorphic, with irg:) 1 for every z є B(0,1). Suppose also that f(0)-0, so f(z)g(2) for some holomorphic function g: B(0,1)C. (a) By applying the Maximum Principle to g on B(0, r) where 0 < r < 1 , deduce that If( S for every 2E (0, 1) . (b) Show also that |f'(0) S1 (c) Show that if lf(z)- for some z B(0,1)\(0), or if If,(0)| = 1 , then there is a...
Let f : [0,1] → R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that 2 - y<== f() - f(y)< € for every 2, Y € [0,1]. The graph of f is the set Gj = {, f(c)): 1 € [0,1]}. : Show that G has measure zero.