# Boolean Algebra

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A 4-basis for Boolean algebra (BA) in terms of join (v), meet (^), and complement (c).

```    x v (y v z) = y v (x v z).  % AJ
x ^ y = c(c(x) v c(y)).     % DM
x v c(x) = y v c(y).        % ONE
(x v c(y)) ^ (x v y) = x.   % CUT
```
Here are proofs of distributivity, modularity, CC, and B1 from the BA 4-basis.
```    otter < BA1.in > BA1.out
otter < BA2.in > BA2.out
otter < BA3.in > BA3.out
```
Independence of the BA 4-basis is open. In particular, we have not been able to find a proof or countermodel of
```    { AJ, DM, CUT } => ONE.
```
The simplest multiequation basis we know of for BA in terms of join and complement is the following, due to C.A. Meredith [13].
```    c(c(x) v y) v x = x.                  % MER_1
c(c(x) v y) v (z v y) = y v (z v x).  % MER_2
```
The Robbins 3-basis for BA (in terms of join and complement) is the following [12].
```    (x v y) v z = x v (y v z).       % AJ2
x v y = y v x.                   % CJ
c(c(x v c(y)) v c(x v y)) = x.   % Robbins
```