190Tutorial7

Tutorial 7

(Mindscape 16, §3.5 of text) Prove that the cardinality of points in a solid cube is the same as the cardinality of points on a line segment.

• With Stephen

The points within a cube have the same cardinality as a line segment because a one-to-one correspondence can be constructed. Take for example a cube with dimensions 0 to .999... in each direction. Any point in the cube can be given a unique point on a line segment from 0-0.999... and vice versa.

Take a point on this cube, say (0.45833,0.08738,0.85586). The unique point this corresponds to can be constructed by taking alternating digits from these coordinates. For example, this point would correspond to the point on the number line at:

x = 0.45833
y = 0.08738
z = 0.85586
0. 408 585 375 338 386

This same process can be applied to any n-dimensional shape.