Made easy Gate mock test -1

Question

A Boolean Function must satisfy the condition f(a,b,c) = f(c,b,a). how many such functions are possible?

Comments

you need to identify the values of triplet (a,b,c) from $2^3$ possible values for which $a_1=c_2$ and $a_2=c_1$ for any two triplets $(a_1,b,c_1)$ and $(a_2,b,c_2)$ because for those values, $f(.)$ will be same according to the question. For example, $f(0,0,1) = f(1,0,0).$

So, out of $2^3$ values, the possible cases are:

$1)$ $\{(0,0,1),(1,0,0)\}$

$2)$ $\{(0,1,1),(1,1,0)\}$

$3)$ $(0,0,0)$

$4)$ $(0,1,0)$

$5)$ $(1,0,1)$

$6)$ $(1,1,1)$

Now, for each $6$ cases, you can assign either $0$ or $1$ as a value of the function and so, total possible boolean functions = $2^6$

---

thanks

Answer 1

Answer: 16

f(a,b,c) = f(c,b,a)   ------------> (given)

i,e.    a = c

We can write, as

 f (a,b,a) or f (c,b,c)   ..........  n = 2

Possible boolean functions = 2^2^n

                                                 = 2^2^2

                                                 = 16

 

Comments

Cool Approach, thanks!!

Answer 2

A Boolean function is a function that can only take on the values of 0 or 1. The number of possible Boolean functions of n variables is 2^(2^n). In the case of f(a,b,c) = f(c,b,a), it means that the function's output does not change when the variables are permuted. This is known as the symmetry property, and it reduces the number of possible Boolean functions.

The number of possible Boolean functions of three variables, a, b, and c, that satisfy the symmetry property is 4. These functions are:

1. Constant functions (f(a,b,c) is always 0 or always 1)
2. Monotone functions (f(a,b,c) is always 1 when at least one of a, b, or c is 1)
3. Linear functions (f(a,b,c) is the exclusive OR of a, b, and c)
4. Bent functions (f(a,b,c) is the exclusive NOR of a, b, and c)

So there are 4 Boolean functions that are possible for f(a,b,c) = f(c,b,a)

Comments

But the correct answer is 64

Answer 3

Here we have 3 variables so max 2$^{3 }$ = 8 min-terms/max-terms are possible. Out of these min-terms there is 2 possibility to choose or reject any min-term but a condition on min-terms is given ,so only those min-terms will be selected which will satisfy the given condition // f(a,b,c) = f(c,b,a)  

Now f(a,b,c) = f(c,a,b) , so only those min-terms will be selected which has a=c

0 0 0 

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

So, only 4 min-terms will be selected. the number of functions = 2$^{4}$ = 16 answer.

PS ; Correct answer to this question is 16 . Made easy later changed the answer from 64 to 16  

Comments

Here, condition is given and based on that we have to make functions. You are doing opposite i.e. taking a particular function and check  given condition holds or not.

If it is given that a=c then your answer will be correct but here it is not given.

---

But sir , Output has to be same . For the example which i took only 4 correct input seq is possible.

 I guess if nothing is mentioned then we should consider the general case .

---

$a=c$ is a particular case not a general case.