Boolean Operations

A table of inputs and result of boolean operations is called the Truth Table

AND

$x$	$y$	$x\wedge y$
0	0	0
0	1	0
1	0	0
1	1	1

OR

$x$	$y$	$x\vee y$
0	0	0
0	1	1
1	0	1
1	1	1
NOT

$x$	$\mathrm{\neg }x$
0	1
1	0

Boolean Expressions

Boolean operations can be combined.

Boolean Functions

Boolean functions have finate input, so a truth table can be listed.

Boolean Identities

Idempotent Law

• $x\wedge x=x$
• $x\vee x=x$

Double Negation Law

• $\mathrm{\neg }(\mathrm{\neg }x)=x$

Commutative Laws

• $x\wedge y=y\wedge x$
• $x\vee y=y\vee x$

Associative Laws

• $x\wedge (y\wedge z)=(x\wedge y)\wedge z$
• $x\vee (y\vee z)=(x\vee y)\vee z$

Distributive Laws

• $x\wedge (y\vee z)=(x\wedge y)\vee (x\wedge z)$
• $x\vee (y\wedge z)=(x\vee y)\wedge (x\vee z)$

De Morgan Laws

• $\mathrm{\neg }(x\wedge y)=\mathrm{\neg }x\vee \mathrm{\neg }y$
• $\mathrm{\neg }(x\vee y)=\mathrm{\neg }x\wedge \mathrm{\neg }y$