# Show how two sets are equinumerous.

The following is a question from a textbook for one of my university classes called computational logic. The class is actually being offered as a philosophy class and since I have no real mathematical background I don't really know how to proceed. any help would be appreciated.

"Show that the set of real numbers is equinumerous with the the set of points on a semicircle from a circle with a diameter of 1."

EDIT: I can now add that the semicircle given in the diagram that accompanied the question is the bottom half of the circle described by $0.25=(x-0.5)^2+(y-0.5)^2$. though I dont know if that really matters.

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I think this picture will make the bijection clear enough. We map the black dots together, where the red line roots from $(0,1)$. By making a detailed drawing, you can get an explicit expression in terms of trigonometric formulas. To be more explicit about how the mapping works, when you make the red line turn with center in $(0,1)$, you connect a point in the semicircle (which, as you ask in the edit, doesn't matter either radius or center, since we can just first map it to the "canonical" unit circle, and then apply this map) with a point in the real line. Because of how we make this connection, you can see points are mappen in a one-one fashion (no two points have the same image) and all points in the circle are mapped to the real line, that is, the map is onto.
$\hspace{2 cm}$
HINT: Assuming that it’s an open semicircle (i.e., one that does not include its endpoints), let the semicircle be the set of points on the circle $x^2+(y-1)^2=1$ with $y$-coordinate less than $1$, and map each point to its projection on the $x$-axis when the centre of projection is the point $\langle 0,1\rangle$.