I am struggling to conceptualize and answer the following question. Consider the relation $R$ that consists of all pairs $(x,y)$ such that $x$ and $y$ are bit strings of length three or more that agree except perhaps in their first three bits. What are the equivalence classes, and how many are there?
I know that $R$ is an equivalence relation, as it is Reflexive (all bits of $x$ are the same as the bits of itself), symmetric (all bits of $x$ (except perhaps the first three) are the same as all bits of y (except perhaps the first three)), and Transitive.
Any help is appreciated!
The elements of a particular equivalence class all share the bits after the 3rd bit in common which means that to every equivalence class can be associated a unique bit-string.
For example: 010 11011 and 011 11011 belong to the same equivalence class, so this equivalence class corresponds to the bit-string 11011 uniquely.
In particular, this means that the equivalence classes of $R$ are bijective with the set of bit-strings itself (so their number is infinite for example).
This is the bijection: $a \mapsto \{000a, 001a, 010a, 011a, 100a, 101a, 110a, 111a\}$