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The problem is stated-

Do the following two sets of points have the same cardinality and if so, establish a bijection:

A line segment of length four and half of the circumference of radius one (including both endpoints).

My reasoning is they do have the same cardinality and my bijection is a picture in which I drew a horizontal line and a semicircle under it (separated by approx. 2cm with the semicircle's open side facing down). At what would be the center of the circle made by the semi-circle I drew a point P. I then drew lines vertical from the original horizontal line to point P. This shows, goes my reasoning, that for every point on the line there is a corresponding point on the semi-circle.

My question is is this drawing enough to show a bijection or do I need to do more?

Thank you for your thoughts.

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    You can simply use the line segment to parameterize the semicircle, as would be done in a vector calculus class. $x = \cos(\pi t/4)$, $y = \sin(\pi t / 4)$.2010-12-15

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I think your lines are not vertical, right? With your construction, there are points on the semi-circle that do not have a corresponding point on the line. But you are quite close. You can 1) modify the construction so that every line from P through the semicircle hits the line (and still have every line from the line segment hit the semi-circle. What does that tell you about the endpoints of each?) Or 2)find another construction that injects the semi-circle into the line, then argue from the Cantor–Bernstein–Schroeder theorem.