A divisor algebra Dn having an even number of Elements is a

A divisor algebra Dn having an even number of Elements is a Boolean algebra Prove this true or given examples of it being false with explanation

Solution

If we define a boolean algebra as having at least two elements, then that algebra has a minimal element (0) and a maximal element (1).

Since each element has a unique complement, 0* = 1 and 1* = 0.

If I add a third element x which is distinct from 0 and 1, it has to have a complement x*.

And of course x* is not equal to 0 or 1, since x isn\'t.

And x* is not equal to x, since if it were we would not have (x v x* = 1). So this algebra has 4 elements.

So no boolean algebra has 3 elements. -- This seems to generalize to: any boolean algebra which has a finite number of elements has an even number of elements. I suppose any finite boolean algebra is isomorphic to a field of sets, where the basis set S has n elements so the field contains 2 to the nsets, which is an even number.