5. Show that S¹same cardinality.{(x, y) = R²|x² + y² = 1} and D² = {(x, y) = R²|x² + y² ≤ 1} have the

To show that the unit circle and the closed unit disk have the same cardinality, establish a…

Answered: - Show that S¹ = {(x, y) = R²|x² + y² =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

5. Show that S¹
same cardinality.
{(x, y) = R²|x² + y² = 1} and D² = {(x, y) = R²|x² + y² ≤ 1} have the

Expert Solution

Step 1: Define function

To show that the unit circle $S to the power of 1$ and the closed unit disk $D squared$ have the same cardinality, establish a bijection between the two sets.

Define a function $f colon S to the power of 1 not stretchy rightwards arrow D squared$ as follows:

$f not stretchy left parenthesis x comma y not stretchy right parenthesis equals not stretchy left parenthesis x comma y not stretchy right parenthesis$

In other words, $f$ takes a point $not stretchy left parenthesis x comma y not stretchy right parenthesis$ on the unit circle $S to the power of 1$ and maps it to the same point $not stretchy left parenthesis x comma y not stretchy right parenthesis$ in the closed unit disk $D squared$ . This function is well-defined because every point on $S to the power of 1$ is also a point in $D squared$ since $S to the power of 1$ is a subset of $D squared$ .