Truth Tables

Any boolean function can be represented with a truth table. A truth table has a line for each possible combination of input values and gives the corresponding outputs. For example,

$A_1$	$A_0$	$B_1$	$B_0$	$C_2$	$C_1$	$C_0$
0	0	0	0	0	0	0
0	0	0	1	0	0	1
0	0	1	0	0	1	0
0	0	1	1	0	1	1
0	1	0	0	0	0	1
0	1	0	1	0	1	0
0	1	1	0	0	1	1
0	1	1	1	1	0	0
1	0	0	0	0	1	0
1	0	0	1	0	1	1
1	0	1	0	1	0	0
1	0	1	1	1	0	1
1	1	0	0	0	1	1
1	1	0	1	1	0	0
1	1	1	0	1	0	1
1	1	1	1	1	1	0

This table represents a function inputing two numbers $A$ and $B$ and outputing the number $C=A+B$. A truth table with $n$ inputs and one output has $2^n$ rows and there are $2^{2^n}$ such truth tables.

Note that the large number of truth tables that it will be impossible to concisely describe some functions - whatever the representation, most functions will have size $\Omega(2^n)$. On the other hand, most functions we are interested in will have small descriptions of some sort.