Question

➲ ǫᴜᴇsᴛɪᴏɴ:⍰

ᴛʜᴇʀᴇ ᴀʀᴇ ɪɴғɪɴɪᴛᴇ ʙʟᴀᴄᴋ ᴀɴᴅ ᴡʜɪᴛᴇ ᴅᴏᴛs ᴏɴ ᴀ ᴘʟᴀɴᴇ. ᴘʀᴏᴠᴇ ᴛʜᴀᴛ ᴛʜᴇ ᴅɪsᴛᴀɴᴄᴇ ʙᴇᴛᴡᴇᴇɴ ᴏɴᴇ ʙʟᴀᴄᴋ ᴅᴏᴛ ᴀɴᴅ ᴏɴᴇ ᴡʜɪᴛᴇ ᴅᴏᴛ ɪs ᴏɴᴇ ᴜɴɪᴛ.✌️​​

Answer 1

Question:-

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

Answer:-

☆Assuming every point of the plane is either white or black.

♧Here's a "constructive" way to find two points of opposite color at distance 1.

Since, there are both white and black points, the

infimum "r" of the distances between white and black points is well defined.

If r > 0, then there exist a black-white pair at distance "d" with r\leqslant d < 2r, and the mid point between them is at distance d/2 < r of either, so it cannot be black or white by the choice of "r", a contradiction.

So r = 0, and there exist a black-white pair at distance d <

1 of each other.

The circles of radius 1 centered at these points intersects, and pairing the two centers with the

two intersection points one gets at least one

black-white pair at distance 1. (infact one gets two pairs)

Answer 2

Answer:

Hey mate❤

If ris greater than zero then there exist a black white pair distance d with r/ d is greater than 2r and the mid point between them is at distance d/2 is greater than r either so it cannot be black or white by the choice of "r", a contradiction.

So r =0 and their exist a black white pair at 1 is greater than d of each other.

Hope it helps you...