TypeShapeBoolean algebraTruth table
AND$$A·B$$
Input Output  
ABA and B
000
010
100
111
OR$$A+B$$
InputOutput
ABA or B
000
011
101
111
NOT$$\ov A$$
InputOutput
Anot A
01
10
NAND$$\ov{A·B}$$
InputOutput
ABA nand B
001
011
101
110
NOR$$\ov{A+B}$$
InputOutput
ABA nor B
001
010
100
110
XOR$$A⊕B$$
InputOutput
ABA xor B
000
011
101
110
XNOR$$\ov{A⊕B}$$ $$A⊙B$$
InputOutput
ABA xnor B
001
010
100
111


Boolean Algebra Theorems

AND, OR are associative: $$A·B·C=(A·B)·C=A·(B·C)$$ $$A+B+C=(A+B)+C=A+(B+C)$$ AND and OR are commutative: $$A·B=B·A$$ $$A+B=B+A$$ Distributive property: $$A+B·C=(A+B)·(A+C)$$ $$A·(B+C)=A·B+A·C$$ DeMorgan's Theorems: $$\ov{A+B}=\ov{A}·\ov{B}$$ $$\ov{A·B}=\ov{A}+\ov{B}$$ Single Variable Theorems: $$A·A=A,\; A+A=A$$ $$A+\ov{A}=1,\; A·\ov{A}=0,\; A=\ov{(\ov{A})}$$ More two-variables Theorems: $$A+A·B=A$$ $$A+\ov{A}·B=A+B$$ Identity and Null operations: $$A·1=A,\; A+1=1$$ $$A+0=A,\; A·0=0$$ $$\ov1=0,\; \ov0=1$$