Question

Show that there are $2^n$entries in a truth table for a function with n inputs.

Short answer

Consider the three inputs, n = 3 . The truth table for three inputs is:

x	y	z
0	0	0
0	0	1
0	1	0
0	1	1
1	0	0
1	0	1
1	1	0
1	1	1

The above truth table has 8 entries, thus $2^n=8$ is proved.

Step-by-step solution

Determine the truth table

Truth tables define the values of the output for the all-possible set of inputs. Each logical operation can be defined by the truth table. For the particular input, combination outputs will be specified in each entry of the truth table.

Determine that there are 2n entries in a truth table for a function with n inputs

Consider the three inputs, n = 3 . The truth table for three inputs is:

x	y	z
0	0	0
0	0	1
0	1	0
0	1	1
1	0	0
1	0	1
1	1	0
1	1	1

We know that n = 3,

$\begin{array}{rcl}{2}^n& =& 2^3\\ & =& 2\times 2\times 2\end{array}$

Then $$\begin{gathered} =4\times 2 \\ =8 \end{gathered}$$

The above truth table has 8 entries.

Therefore, it is shown that there are $2^n$ entries in a truth table for a function with n inputs.