Prove the following for all x belongsto R: x is rational x/5 is rational x -...
Question 4 of the image
Prove that, for all n 1 1 Arrange the following rational numbers in increasing order: (i) x, is a rational number 61/99, 3/5, 17/30, 601/999, 599/1001. g 0 2 Find positive integers r and s such that r/s is equal to the repeating decimal (ii) 2 x5/2. Find an expression for x - 5 involving x,-5, and hence explain (without formal proof) why x, tends to a limit which is not a rational number 0.30024....
If r and s are rational numbers, prove that r + s is a rational number.
1. Let f:R → R be the function defined as: 32 0 if x is rational if x is irrational Prove that lim -70 f(x) = 0. Prove that limc f(x) does not exist for every real number c + 0. 2. Let f:R + R be a continuous function such that f(0) = 0 and f(2) = 0. Prove that there exists a real number c such that f(c+1) = f(c). 3 Let f. (a,b) R be a function...
5. Prove that v6 is not rational (it is irrational)
5. Prove that v6 is not rational (it is irrational)
(5) Prove that for Anosov map induce on the 2D-torus T2 by using x1 = 2x + y, yi = X + y, all (x, y) of coordinates of rational numbers are on periodic orbits. w transcribed image text
(5) Prove that for Anosov map induce on the 2D-torus T2 by using x1 = 2x + y, yi = X + y, all (x, y) of coordinates of rational numbers are on periodic orbits. w transcribed image text
2. [14 marks] Rational Numbers The rational numbers, usually denoted Q are the set {n E R 3p, q ZAq&0An= Note that we've relaxed the requirement from class that gcd(p, q) = 1. (a) Prove that the sum of two rational numbers is also a rational number (b) Prove that the product of two rational numbers is also a rational number (c) Suppose f R R and f(x)= x2 +x + 1. Show that Vx e R xe Qf(x) Q...
5. (a) (5 points) Let R F[x] for a field F. Let f, g E R be nonzero. Prove that (f(x)) = (g(x)) if and only if g(x) = af(x) for some constant a E F. (b) (5 points) Let R be any ring. Prove that the nilradical Vo is contained in the intersection of all prime ideals.
# 7 please
6. Prove that if x is rational and y is irrational, then 2 +y is irrational. 7. Prove that if x, y € R+ such that Ty # #4, then x + y.
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...
5. (5 pts) a. Write the equation for a rational function r(x) that has a vertical asymptote at x = 8, a horizontal asymptote at y=1, and a y-intercept at (0, -1). #5a: b. Find the x-intercept for your function.