Is 3 bits difference enough to map distinct values toggling a single bit in a binary number?

2018-10-20 13:34:57

Assume that one can only alter a single bit in a binary number set. As in:

00000000

can be converted to one of the rows in:

00000001

00000010

00000100

00001000

00010000

00100000

01000000

10000000

If one has the set 00000000, 00000111 all 16 conversions(mappings) will be distinct. How to prove, or disprove, that:

If any of the numbers in the set has at least 3-bits of difference, the values they map to will be distinct.

?