Let R be the set of points on the unit circle with rational coordinates, i.e., R consists of those pairs (q, r) of rational numberswith q2 + r2 = 1. Show that R is denumerable. [Hint: Consider a line through (−1, 0) with rational slope, and work out where it intersects the unit circle.]

Let R be the set of points on the unit circle with rational coordinates, i.e., R consists of those pairs (q, r) of rational numberswith q2 + r2 = 1. Show that R is denumerable. [Hint: Consider a line through (−1, 0) with rational slope, and work out where it intersects the unit circle.]

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.3: Lines

Problem 70E

See similar textbooks

Related questions

Q: For this problem, let f: RR be a function. We say that a non-empty set UCR is ajar if Vu €U, 3r >0…

A: A non empty set U subset of R is called ajar if for all u belongs to U there exists a r>0 such…

Q: Can you answer these four questions, please? Find the equation of the line joining the points…

Q: Find the vector determined by 18 || = B = {[.]. 9 6 12 3 9 2 -8 " where B = 3 [-]

Q: A linear second-order non-homogeneous equation models this scenario: People falling at a height of…

A: The objective of this question is to convert a second-order non-homogeneous differential equation…

Q: Let F = (6x²y + 2y³ +3e)i + (7e +54x) 3. Consider the line integral of Faround the circle of radius…

Q: 2x-3y+z=6 2y + 4z = 12 x-y+2z=1

A: The system of linear equationsto solve for x, y, and z.

Q: Exercise 10.1.1 Consider the following functions T: R³ → R². For each of these functions T, show…

A: (.) Given, be a function . We have to show that given functions is not linear transformation.(.)…

Q: Compute the work done by the force 7 = (sin(x + y), xy, x²z) in moving an object along the…

A: Given force The object is moving along the straight line and from

Q: Find R 4x dA, where R is a region enclosed by 4x − y = 0, 4x − y = 2, x + y = 0 and x + y = 4. Use…

Q: g) f(t)= (1-3)² U₂ (1)

Q: Find the first four nonzero terms in a power series expansion about x = 0 for a general solution to…

Q: Problem 5. Prove that the space X is Hausdorff if and only if the set ▲ = {(x,x) : x ¤ X} is closed…

Q: Using residues, where a > 0 evaluate the integral L²+ 3x sin ax (x²+4)(x² + 16) 8 dx

A: Using residues, where a > 0 We have to evaluate given integral.

Q: Compute u • u, vou, and u u using the vectors u = 3 and v= 7

Q: Let B = {b₁,b₂} and C = = {C₁, C₂}. Find [v] and [v]. 98 6+ 10 [v]. [v] 8 9 -8 -7 -6 = b2 c2 bl Lo…

Q: A linear second-order non-homogeneous equation models this scenario: People falling at a height of…

A: The objective of the question is to represent the given scenario as an initial value problem in…

Q: x[n] is shown in the figure below. Sketch the sequences x₁ [n] and x₂ [n] specified as: x₁ [n] =…

Q: Find the change of basis matrix [B]. 8 8 B = { [] [] }; * = { [] []} C 9

A: Here we will express each vector in the basic B (1st basis) as a linear combination of vectors in…

Q: Σ n=1 1 η3

Q: Prove by induction that Vn N 72 n(n+1)(2n - 2) Σ(i²-i) = 6 i=1

Q: Wouldn't the values of r be something else for the integral? The shaded region starts at the circle…

A: The region shown in the figure.

Q: Question 1. The inverse Laplace transform of 88²-628-6 (s+6) (s2-88+25) is: peat+qest cos(wt) + ret…

Q: Let V and W be vector spaces and A : V → W be a linear map. (a)The kernel of a linear map f : V → W…

Q: From the following function, graph in the indicated Interval f(x) = A 2 1-2x + 2π -< X< TT X</ 2 <X<…

A: Given Information:The function is provided as To find:a) The value of as area under the curve.b)…

Q: Consider a plane curve defined by the parametric equations : x(t)=acos^3 (t) and y(t)=asin^3 (t)…

A: The curvature κ(t) of a plane curve defined by parametric equations x(t) and y(t) is given…

Q: List the eigenvalues of the matrix A₁, A₂= -1,1 Find an eigenvector ₁ for A₁ = -1: 1 3 Find an…

Q: 2. Determine the Laplace transform of the given function a) f(t) = e³¹t4 b) f(t

A: Laplace transform

Q: 10 Consider the function f(x) = 5x¹0 + 7x³ – 6x¹ – 4. Enter an antiderivative of f(x). Do not enter…

A: The antiderivative of a function is a function such that its derivative equals the original…

Q: bits are needed.

Q: Exercise 10.4.2 Let B = in R². Find [x]B. -1 MGA 0 2 be a basis of R3 and let x = 41 -1 be a vector

A: Goven that be the basis of and

Q: X 1+x² List the elements of the requested image or preimage. No set notation needed. Let q be the…

Q: (d) Find a unit eigenvector, v2, for eigenvalue A₂. (e) Construct a matrix P such that A = PDP-¹ = P…

Q: A stone is thrown straight up from the edge of a roof, 950 feet above the ground, at a speed of 20…

Q: Let B = {b₁,b2} and = {C₁, C₂}. Find [v] and [v]. B 10 -9 [v] Ⓡ [v] 8 -7 -6 -5 -4 -3 -2 -1 6- 5 4 3…

Q: Do not Claim ( For all real numbers a, b, c, and do if a = (b +c+d)/3, then either (2b + c) ≤3a, or…

Q: What vector v is given by the coordinate vector 3 [1] 6 B={[-3 8 7], [7 1 6], [7 -4 9]}. ?

Q: The graph below is of a function g(x). -3 -2 Use the graph to estimate Enter a number, 7 -1 (a) a…

A: The graph below is of a function .By using the graph to estimate , by using six sub-intervals of…

Q: Express the following integral as a double integral in polar coordinates 1 Suvaldy x² + y² dy dr

Q: Exercise 10.1.1 Consider the following functions T: R³ → R². For each of these functions T, show…

A: Note: we will be solving part(c) only as mentioned.Consider the function defined as follows..We…

Q: 2 y² - x² = 9 a) Sketch the graph of the equations in three dimensions and identify the surfaces.

Q: Implement an eﬀective Aggregate Planning Process using Chase Strategy, Level Strategy, and Mixed…

A: In tackling the challenge of effective Aggregate Planning, three prominent strategies are employed:…

Q: 2. Suppose that c₁,C2,C3,... is a sequence defined as follows c₁=0, c₂=-10, c3 = 10³ and Ck=11ck-3…

Q: Everything we've done with systems of 2 linear, constant co- efficient, homogeneous differential…

A: Let let the charecteristic equation of A is (or) ------- (1) where…

Q: Approximate projíu and projów. -8 -7 -6 -5 -4 -3 -2 -1 projwu projuw vector u vector w -5 Each entry…

Q: If A = (aij) = M3 (K) for a field K then the characteristic polynomial of A has the form p₁(x) = -…

Q: Given ƒ"(x) = 7x +5 and ƒ'(0) = −2 and ƒ(0) = 1. Find f'(x) and find ƒ(4) =

A: Given,this is the required answer of the given question

Q: = ²√x = by identifying the domain and any symmetries, finding the derivatives y' and y", finding the…

Q: 2. Suppose that co,c1,c2,... is a sequence defined as follows: co=1, C₁=4 Ck=2ck-1-Ck-2 for every…

A: Suppose that is a sequence defined as follows :…

Q: For which of the following matrices is [1 2] TT [23] 0 TO 2110 H an eigenvector with a nonzero…

Q: 4. Solve the following boundary value and initial value problem for wave equation Utt-c²Uzz = 0,0 0…

A: It is given that where and . Here, the initial and boundary conditions are:We have to find the…

Question

Let R be the set of points on the unit circle with rational coordinates, i.e., R consists of those pairs (q, r) of rational numberswith q² + r² = 1. Show that R is denumerable. [Hint: Consider a line through (−1, 0) with rational slope, and work out where it intersects the unit circle.]

Expert Solution