UGC NET CSE | June 2019 | Part 2 | Question: 13

Question

How many different Boolean functions of degree $n$ are the

• A. $2^{2^n}$
• B. $(2^2)^n$
• C. $2^{2^n} -1$
• D. $2^n$ 

Comments

Answer $1)$

If we do $2$ to $n$ mapping, n boolean element can make set of $2^{n}$ elements.

Now for mapping of $2$ to $2^{n}$, boolean function would be $2^{2^{n}}$

Answer 1

$A$	$B$	$f_0$	$f_1$	$f_2$	$f_3$	$f_4$	$f_5$	$f_6$	$f_7$	$f_8$	$f_9$	$f_{10}$	$f_{11}$	$f_{12}$	$f_{13}$	$f_{14}$	$f_{15}$
0	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1
0	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
1	0	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
1	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1

As we can see from the above table ,

Using $n$ boolean variables (here $n=2$ $A,B$ are boolean variables),

We can create $2^n$ combinations ($00,01,10,11$)

and using those $2^n$ combinations we can create $2^{2^n}$ functions ($16$ functions = $f_0,f_1,....f_{15}$)

$\therefore$ Option $1.$ is correct.