(1) A .0 = 0 {\displaystyle A.0=0\,}
(2) A .1 = A {\displaystyle A.1=A\,}
(3) A . A = A {\displaystyle A.A=A\,}
(4) A . A ¯ = 0 {\displaystyle A.{\bar {A}}=0\,}
(5) A + 0 = A {\displaystyle A+0=A\,}
(6) A + 1 = 1 {\displaystyle A+1=1\,}
(7) A + A = A {\displaystyle A+A=A\,}
(8) A + A ¯ = 1 {\displaystyle A+{\bar {A}}=1\,}
(9) A + B = B + A {\displaystyle A+B=B+A\,}
(10) A . B = B . A {\displaystyle A.B=B.A\,}
(11) A + ( B + C ) = ( A + B ) + C = A + B + C {\displaystyle A+(B+C)=(A+B)+C=A+B+C\,}
(12) A ( B . C ) = ( A . B ) . C = A . B . C {\displaystyle A(B.C)=(A.B).C=A.B.C\,}
(13a) A ( B + C ) = A B + A C {\displaystyle A(B+C)=AB+AC\,}
(13b) ( A + B ) . ( C + D ) = A C + A D + B C + B D {\displaystyle (A+B).(C+D)=AC+AD+BC+BD\,}
(14) A + ( A B ) = A {\displaystyle A+(AB)=A\,}
(15a) A + ( A ¯ B ) = A + B {\displaystyle A+({\bar {A}}B)=A+B\,}
(15b) A ¯ + ( A B ) = A ¯ + B {\displaystyle {\bar {A}}+(AB)={\bar {A}}+B\,}
(a) A B ¯ C + A ¯ B ¯ C ¯ + A B ¯ C ¯ = . . . {\displaystyle A{\bar {B}}C+{\bar {A}}{\bar {B}}{\bar {C}}+A{\bar {B}}{\bar {C}}=...\,}
(b) A B ¯ D + A B ¯ D ¯ + A ¯ B D = . . . {\displaystyle A{\bar {B}}D+A{\bar {B}}{\bar {D}}+{\bar {A}}BD=...\,}
(c) ( A ¯ + B ) . ( A + B ) = . . . {\displaystyle ({\bar {A}}+B).(A+B)=...\,}
(d) A C D + A ¯ B C D = . . . {\displaystyle ACD+{\bar {A}}BCD=...\,}
(e) S = A C ¯ + A B C ¯ = . . . {\displaystyle S=A{\bar {C}}+AB{\bar {C}}=...\,}
(f) S = A ¯ B ¯ C D ¯ + A ¯ B ¯ C ¯ D ¯ = . . . {\displaystyle S={\bar {A}}{\bar {B}}C{\bar {D}}+{\bar {A}}{\bar {B}}{\bar {C}}{\bar {D}}=...\,}
(g) S = A ¯ D + A B D = . . . {\displaystyle S={\bar {A}}D+ABD=...\,}
(16) ( A + B ¯ ) = A ¯ . B ¯ {\displaystyle ({\overline {A+B}})={\bar {A}}.{\bar {B}}\,}
(17) ( A . B ¯ ) = A ¯ + B ¯ {\displaystyle ({\overline {A.B}})={\bar {A}}+{\bar {B}}\,}
S = A ¯ B ¯ C D ¯ + A ¯ B C D + A B ¯ C ¯ D ¯ + A B ¯ C D ¯ + A B C ¯ D + A B C D {\displaystyle S={\bar {A}}{\bar {B}}C{\bar {D}}+{\bar {A}}BCD+A{\bar {B}}{\bar {C}}{\bar {D}}+A{\bar {B}}C{\bar {D}}+AB{\bar {C}}D+ABCD}
S = A ¯ B ¯ C ¯ D ¯ + A ¯ B ¯ C ¯ D + A ¯ B ¯ C D + A ¯ B C ¯ D ¯ + A ¯ B C ¯ D + A ¯ B C D ¯ + A B ¯ C ¯ D ¯ + A B ¯ C ¯ D + A B ¯ C D + A B C ¯ D ¯ + A B C D ¯ {\displaystyle S={\bar {A}}{\bar {B}}{\bar {C}}{\bar {D}}+{\bar {A}}{\bar {B}}{\bar {C}}D+{\bar {A}}{\bar {B}}CD+{\bar {A}}B{\bar {C}}{\bar {D}}+{\bar {A}}B{\bar {C}}D+{\bar {A}}BC{\bar {D}}+A{\bar {B}}{\bar {C}}{\bar {D}}+A{\bar {B}}{\bar {C}}D+A{\bar {B}}CD+AB{\bar {C}}{\bar {D}}+ABC{\bar {D}}}