Bit (abstract)

An abstract bit is an element of the set [boolean $\mathbb B$], which has two elements. An abstract bit is to $\mathbb B$ as a number is to [natural_number $\mathbb N$]. The set $\mathbb N$ contains a bunch of "numbers", and these numbers sometimes have many different labels, such as "three" and "3". Similarly, the set $\mathbb B$ contains a pair of "bits", which sometimes have many different labels. The abstract bits are sometimes labeled 0 and 1; true and false; yes and no; or + and -. The labels don't matter; the point is that the set $\mathbb B$ contains two elements, and we call those elements (abstract) bits.