Type | Shape | Boolean algebra | Truth table |
AND |  | $$A·B$$ |
Input | Output |
A | B | A and B |
0 | 0 | 0 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 1 |
|
OR |  | $$A+B$$ |
Input | Output |
A | B | A or B |
0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 1 |
|
NOT |  | $$\ov A$$ |
|
NAND |  | $$\ov{A·B}$$ |
Input | Output |
A | B | A nand B |
0 | 0 | 1 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 0 |
|
NOR |  | $$\ov{A+B}$$ |
Input | Output |
A | B | A nor B |
0 | 0 | 1 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 0 |
|
XOR |  | $$A⊕B$$ |
Input | Output |
A | B | A xor B |
0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 0 |
|
XNOR |  | $$\ov{A⊕B}$$ $$A⊙B$$ |
Input | Output |
A | B | A xnor B |
0 | 0 | 1 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 1 |
|
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:
$$A·1=A,\; A+1=1$$ $$A+0=A,\; A·0=0$$ $$\ov1=0,\; \ov0=1$$