Cardinality of Line Segment & Half Circumference: Bijection Established

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In summary, the question asks whether a line segment of 4 units and half of a circumference of radius 1 have the same cardinality. The solution proposed involves placing one figure inside the other and establishing a bijection by radiating lines from the center of each to the edge of the larger circle. However, the question only asks for a bijection and not for an explanation of why the figures have the same cardinality, so simply writing out an equation would suffice.
  • #1
MikeDietrich
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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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  • #2
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:​
 

Related to Cardinality of Line Segment & Half Circumference: Bijection Established

What is the cardinality of a line segment?

The cardinality of a line segment is infinite. This means that there are an infinite number of points on a line segment, and it cannot be counted or measured in a traditional sense.

What is the half circumference of a circle?

The half circumference of a circle is equal to half of the total circumference of the circle. This can be calculated using the formula C/2 = πr, where C is the circumference and r is the radius of the circle.

How is a bijection established between the cardinality of a line segment and the half circumference of a circle?

A bijection is established between the cardinality of a line segment and the half circumference of a circle by mapping each point on the line segment to a unique point on the half circumference. This can be visualized by imagining a straight line being wrapped around a circle, with each point on the line corresponding to a point on the half circumference.

Can a bijection be established between any line segment and any half circumference?

Yes, a bijection can be established between any line segment and any half circumference, as long as the line segment and half circumference have the same cardinality. This means that the line segment and half circumference must have an equal number of points.

How does the concept of bijection relate to the cardinality of sets?

Bijection is a one-to-one correspondence between elements of two sets. This means that for every element in one set, there is a corresponding element in the other set, and vice versa. In the case of the cardinality of a line segment and half circumference, a bijection can be established because both sets have an infinite number of elements, despite their different visual representations.

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