Question

There are infinite black and white dots on a plane. Prove that the distance between one black dot and one white dot is one unit.

Answer 1

Theree are two ways to look at this

Answer 2

1) all the points on the plane are either black or white, and there are infinite black and white points (this is probably what your instructor meant)

2) there exist infinite black and white points on the plane (Englis)

Answer 3

I cant type rn

Answer 4

Case 1)
Assume that there is no black point which is at a dist of one unit from a white pt

Pick a black pt and make a circle of radius 1 around it
From our assumption, it's trivial to say that all the points on the circle will also be blackk

Every point inside the circle will have one point on the circumference which Is at a distance of one unit from it. So all the points inside have to be black as well

So essentially you have a black circular patch. You can pick another point on this circle as the centre and repeat the same process. You'll end up with a plane filled with all black points and 0 white points, which Is a contradiction since we have infinite white issues (given)

So the assumption is not possible. Theree has to be one white pt at a dist of 1 unit from a black pt

Answer 5

Case2)
Consider this example-

All(infinite) the black points lie to the left of x=0
All(infinite) the white points lie to the right of x=2

The plane isn't filled with black and white pts, but you have infinite black and white points with no black point at a dist of one unit from a white pt

Answer 6

^^infinite white points*