These are some of the laws of boolean logic they tought me in
school. They can be used to simplify complex boolean equations.

The dot (.) means logical and, the + logical or, and the ! is
inversion. Forgive me if I get the names of the laws wrong, but I
went to a dutch school, so I just translated them.

AND-Laws OR-Laws
A = !!A inversion A = !!A
A . !A = 0 1st basic law A + !A = 1
A . A = A 2nd basic law A + A = 1
A . 1 = A 3rd basic law A + 1 = 1
A . 0 = 0 4th basic law A + 0 = A
A . B = B . A commutative A + B = B + A
A . (A + B) = A true absorption A + (A . B) = A
A . (!A + B) = A . B false absorption A + (!A . B) = A + B
(A + B) . (A + C) = A + (B . C) distribution (A . B) + (A . C) = A . (B + C)
A . (B . C) = B . (A . C) = C . (A . B) associative A + (B + C) = B + (A + C) = C + (A + B)
!(A . B) = !A + !B Morgan !(A + B) = !A . !B