We wish to create a partition of the Cartesian plane. Determine if the following sets are...

We wish to create a partition of the Cartesian plane. Determine if the following sets are a partition and explain why or why not.
1. The set of all circles with radius 3 units and varying centers.
2. The set of all circles with varying radii and centered at the point (0,0) (called the origin). Note the origin is a circle with radius zero.
3. The set of all vertical lines.

Expert Solution

a) Two circles of radii 3 with centers (0,0) and (1,2) is given below

These are different sets but they are not disjoint. This means the given sets don't partition the plane

b) Each circle is uniquely determined by its radius

Every element (a, b) belongs to the set which is a circle with center at origin and radius

Any two circles of different radii will never touch as this would mean the two radii are equal at some point which is impossible. So the sets are disjoint and cover the whole plane. Meaning the given sets form a partition

c) Each point (a, b) belongs to the vertical line x=a. As all vertical lines are parallel to each other, any two such sets just be disjoint.

Therefore we again get a partition.

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Colby Messinger answered 2 years ago

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