Refinement on partition of bit strings

Question

A partition is $P_1$ called refinement of the partition $P_2$, if every set in $P_1$ is a subset of one of the sets in $P_2$.

The partition of the set of bit strings of length 16 formed by the equivalence classes of bit strings that agree on the last eight-bits is a refinement of the partition formed from the equivalence classes of bit strings that agree on the last four bits.

How strings that agree on the last eight bits be a subset of the bit strings that agree on the last four bits?

Answer 1

If $E_1, E_2 \subseteq A^2$ are both equivalence relations on $A$, and $E_1 \subseteq E_2$, then $A/E_1$ is a refinement of $A/E_2$ ($\Leftarrow$ for any $a\in A$, $a/E_1 = \{b : b\in A, (a,b)\in E_1\}\subseteq\{b : b\in A, (a,b)\in E_2\} = a/E_2$).

And if two bit strings agree on the last $8$ bits, they agree on the last $4$ bits.

Answer 2

Consider the set ${\cal S}$ of strings of length eight in which the last four digits match. In some of these cases the first four digits match, but in other cases the first four do not match.

Hence the set in which the first four digits match is a subset of ${\cal S}$.