Boolean Algebra

Boolean Operators §

The AND operator ($\cdot$) can be defined by the following truth table:

A	B	Q
0	0	0
0	1	0
1	0	0
1	1	1

The OR operator ($+$) can be defined by the following truth table:

A	B	Q
0	0	0
0	1	1
1	0	1
1	1	1

The XOR (eXclusive OR) operator ($\oplus$) can be defined by the following truth table:

A	B	Q
0	0	0
0	1	1
1	0	1
1	1	0

The unary NOT operator ($\overline{}$) can be defined by the following truth table:

A	Q
0	1
1	0

Composite Operations §

There are some very common operations that are best defined as formulae involving other operations. For example:

• A NOR B is $\overline{A+B}$
• A NAND B is $\overline{A\cdot B}$
• A XNOR B is $\overline{A\oplus B}$

De Morgan’s Laws §

De Morgan’s Laws state that:

$\overline{(A\cdot B)}=\bar{A}+\bar{B}$ and $\overline{(A+B)}=\bar{A}\cdot \bar{B}$

Associativity and Commutativity §

AND, OR and XOR are all both associative and commutative.