Chapter 14.1, Problem 21WE

To determine

To explain: The reason that the mapping $M$ does not preserve distance.

Explanation of Solution

Given information: It is given that mapping $M$ maps points $A$ and $B$ to the same image point.

A mapping from set A to set B is called a one-to-one mapping if every number of B has exactly one pre-image in A .

Now consider $A^{\prime }$ is the image of the point A and $B^{\prime }$ is the image of point B .

  $M:A\to A^{\prime }$

And

  $M:B\to B^{\prime }$

As the points are at different locations therefore the length of AB will be greater than the zero.

  $AB>0$

But as mapping M maps points A and B to the same image point, the length of the line $A^{\prime }{B}^{\prime }$ will be equal to zero.

  $A^{\prime }{B}^{\prime }=0$

Therefore, the mapping does not preserve distance.