Cardinality of Line Segment & Half Circumference: Bijection Established

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SUMMARY

The discussion centers on establishing the cardinality between a line segment of 4 units and half of a circumference of radius 1. Participants confirm that these two geometric figures have the same cardinality and emphasize the need to establish a bijection. A bijection can be demonstrated by mapping points from the line segment to the half circumference using a defined equation, thus providing a complete explanation of their equivalence in cardinality.

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  • Understanding of cardinality in set theory
  • Familiarity with bijections and their properties
  • Basic knowledge of geometric figures, specifically line segments and circumferences
  • Ability to manipulate and visualize geometric shapes
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  • Research the concept of cardinality in set theory
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Mathematicians, students studying set theory, educators teaching geometry, and anyone interested in the properties of geometric figures and their cardinalities.

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Homework Statement
Do the following have the same cardinality? If so, establish a bijection and if not explain why.

A line segment of 4 units and half of a circumference of radius 1 (including both endpoints).


The attempt at a solution

So my thought is that if I manipulate the shape of these two figures I have two circles of different sizes and if I place one inside the other then I can show they have the SAME cardinality by radiating lines from the center of each to the edge of the large circle.

First of all, I am not even sure if this is appropriate/accurate to do and then I do not know how to establish a bijection (aside the picture).

Any thoughts or suggestions? Thank you in advance!
 
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Hi Mike! :smile:

You're both being too analytical and not reading the question carefully enough :wink: …
MikeDietrich said:
Do the following have the same cardinality? If so, establish a bijection and if not explain why.

The question does not ask you to explain why, in the case that they do have the same cardinality …

it only asks for a bijection (which of course is a complete explanation in itself anyway).

So just write out an equation. :smile:​
 

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