Theorems
in pairs
basic theorems
Th1. A + A = A
Th2. A * A = A
Th3. A + 0 = A
Th4. A * 1 = A
Th5. A * 0 = 0
Th6. A + 1 = 1
-> DeMorgan 's theorems
Th7. (A + B)'= A'* B'
Th8. A'* B = A'+ B'
Th9. A + A * B = A
Th10. A * (A + B) = A
Th11. A + A'* B = A + B
Th12. A'* (A + B') = A'* B'
Th13. A * B + A * B' = A
Th14. (A'+ B') * (A'+ B) = A'
Th15. A + A' = 1
Th16. A * A' = 0
Th.1
A + A = A
A | B | A OR B
--------------
0 | 0 | 0 <
0 | 1 | 1
1 | 0 | 1
1 | 1 | 1 <
Th.2
A * A = A
A | B | A AND B
---------------
0 | 0 | 0 <
0 | 1 | 0
1 | 0 | 0
1 | 1 | 1 <
Th.3
A + 0 = A
A | B | A OR B
--------------
0 | 0 | 0 <
0 | 1 | 1
1 | 0 | 1 <
1 | 1 | 1
Th.4
A * 1 = A
A | B | A AND B
---------------
0 | 0 | 0
0 | 1 | 0 <
1 | 0 | 0
1 | 1 | 1 <
Th.5
A * 0 = 0
A | B | A AND B
---------------
0 | 0 | 0 <
0 | 1 | 0
1 | 0 | 0 <
1 | 1 | 1
Th.6
A + 1 = 1
A | B | A OR B
--------------
0 | 0 | 0
0 | 1 | 1 <
1 | 0 | 1
1 | 1 | 1 <
-> DeMorgan 's theorems
Th7.
(A + B)' = A'* B'
A | B | A+B | (A+B)'| A'| B'| A'B'
----------------------------------
0 | 0 | 0 | 1 | 1 | 1 | 1
0 | 1 | 1 | 0 | 1 | 0 | 0
1 | 0 | 1 | 0 | 0 | 1 | 0
1 | 1 | 1 | 0 | 0 | 0 | 0
^ ^
(0 + 0)' = 0'* 0'
} => (A + 0)' = A'* 0'
(1 + 0)' = 1'* 0' A' = A'* 0'
A' = A'* 1
A' = A'
(0 + 1)' = 0'* 1'
} => (A + 1)' = A'* 1'
(1 + 1)' = 1'* 1' 1' = A'* 1'
0 = A'* 0
0 = 0
Th.8
(A * B)' = A'+ B'
A | B | A*B | (A*B)'| A'| B'| A'+ B'
------------------------------------
0 | 0 | 0 | 1 | 1 | 1 | 1
0 | 1 | 0 | 1 | 1 | 0 | 1
1 | 0 | 0 | 1 | 0 | 1 | 1
1 | 1 | 1 | 0 | 0 | 0 | 0
^ ^
(0 * 0)'= 0'+ 0'
} => (A * 0)'= A'+ 0'
(1 * 0)'= 1'+ 0' 0' = A'+ 1
1 = 1
(0 * 1)'= 0'+ 1'
} => (A * 1)'= A'+ 1'
(1 * 1)'= 1'+ 1' A' = A'+ 0
A' = A'
Th.9
A + A * B = A
A | B | A*B | A + AB
--------------------
0 | 0 | 0 | 0
0 | 1 | 0 | 0
1 | 0 | 0 | 1
1 | 1 | 1 | 1
^ ^
0 + 0 * 0 = 0
} => A + A * 0 = A
1 + 1 * 0 = 1 A + 0 = A
A = A
0 + 0 * 1 = 0
} => A + A * 1 = A
1 + 1 * 1 = 1 A + A = A
A = A
----
Th.10
A * (A + B) = A
A | B | A+B | A(A+B)
--------------------
0 | 0 | 0 | 0
0 | 1 | 1 | 0
1 | 0 | 1 | 1
1 | 1 | 1 | 1
^ ^
0 * (0 + 0) = 0
} => A * (A + 0) = A
1 * (1 + 0) = 1 A * A = A
A = A
0 * (0 + 1) = 0
} => A * (A + 1) = A
1 * (1 + 1) = 1 A * 1 = A
A = A
----
Th.11
A + A'* B = A + B
A'| B | A | A'B | A+A'B | A+B
-----------------------------
1 | 0 | 0 | 0 | 0 | 0
1 | 1 | 0 | 1 | 1 | 1
0 | 0 | 1 | 0 | 1 | 1
0 | 1 | 1 | 0 | 1 | 1
^ ^
0 + 0'* 0 = 0 + 0
} => A + A'* 0 = A + 0
1 + 1'* 0 = 1 + 0 A + 0 = A
A = A
0 + 0'* 1 = 0 + 1
} => A + A'* 1 = A + 1
1 + 1'* 1 = 1 + 1 A + A' = 1
----
Th.12
A' * (A + B') = A'B'
A | A'| B'| A+B'|A'*(A+B')| A'B'
--------------------------------
0 | 1 | 1 | 1 | 1 | 1
0 | 1 | 0 | 0 | 0 | 0
1 | 0 | 1 | 1 | 0 | 0
1 | 0 | 0 | 1 | 0 | 0
^ ^
1' * (1 + 0') = 1' * 0'
} =>
0' * (0 + 0') = 0' * 0'
A' *(A + 0') = A'* 0'
A' *(A + 1 ) = A'* 1
A' * 1 = A'* 1
A' = A'
0' * (0 + 1') = 0' * 1'
} =>
1' * (1 + 1') = 1' * 1'
A' *(A + 1') = A'* 1'
A' *(A + 0 ) = A'* 0
A' * A = 0
Th.13
A * B + A * B' = A
A | B | B'| A*B | A*B'| AB+AB'
------------------------------
0 | 0 | 1 | 0 | 0 | 0
0 | 1 | 0 | 0 | 0 | 0
1 | 0 | 1 | 0 | 1 | 1
1 | 1 | 0 | 1 | 0 | 1
^ ^
0 * 0 + 0 * 0'= 0
} => A * 0 + A * 0'= A
1 * 0 + 1 * 0'= 1 A * 0 + A * 1 = A
0 + A = A
A = A
0 * 1 + 0 * 1'= 0
} => A * 1 + A * 1'= A
1 * 1 + 1 * 1'= 1 A * 1 + A * 0 = A
A + 0 = A
A = A
----
Th.14
(A' + B') * (A' + B) = A'
A'| B | B'| A'+B' | A'+ B |(A'+B')*(A'+B)
-----------------------------------------
1 | 0 | 1 | 1 | 1 | 1
1 | 1 | 0 | 1 | 1 | 1
0 | 0 | 1 | 1 | 0 | 0
0 | 1 | 0 | 0 | 1 | 0
^ ^
(0' + 0') * (0' + 0) = 0'
} =>
(1' + 0') * (1' + 0) = 1'
(A' + 0') * (A' + 0) = A'
(A' + 1 ) * A' = A'
1 * A' = A'
A' = A'
(0' + 1') * (0' + 1) = 0'
} =>
(1' + 1') * (1' + 1) = 1'
(A' + 1') * (A' + 1) = A'
(A' + 0 ) * 1 = A'
A' * 1 = A'
A' = A'
----
Th.15
A + A' = 1
A | A'| A+A'
------------
0 | 1 | 1 <
1 | 0 | 1 <
1 + 1' = 1 =>
1 + 0 = 1
0 + 0' = 1 =>
0 + 1 = 1
Th.16
A * A' = 0
A | A'| A*A'
------------
0 | 1 | 0 <
1 | 0 | 0 <
1 * 1' = 0
1 * 0 = 0
0 * 0' = 0
0 * 1 = 0