booleanLogic.md

Boolean logic

Fundamental gates

A	NOT
0	1
1	0

A	B	AND		A	B	NAND		A	B	OR		A	B	NOR		A	B	XOR
0	0	0	---	0	0	1	---	0	0	0	---	0	0	1	---	0	0	0
0	1	0	---	0	1	1	---	0	1	1	---	0	1	0	---	0	1	1
1	0	0	---	1	0	1	---	1	0	1	---	1	0	0	---	1	0	1
1	1	1	---	1	1	0	---	1	1	1	---	1	1	0	---	1	1	0

Basic Boolean Algebraic Identities and Properties

ADDITIVE		MULTIPLICATIVE
A + 0 = A		0A = 0
A + 1 = 1		1A = A
A + A = A		AA = A
A + Ā = 1		AĀ = 0
A+B = B+A		AB = BA
A+(B+C) = (A+B)+C	A(B+C) = AB + AC	A(BC) = (AB)C

Boolean rules for semplification

A + AB = A
A + ĀB = A + B
(A + B) (A + C) = A + BC

De Morgan's Theorems

Not(AB) = Not(A) + Not(B)
Not(A + B) = Not(A)Not(B)

We can use De Morgan's laws and boolean arithmetic and rule from semplifications to go from truth tables to logic circuits using Sum of Products for 1 or Product of sums for 0.