# GATE | GATE-CS-2015 (Set 1) | Question 65

Consider the operations
f(X, Y, Z) = X’YZ + XY’ + Y’Z’  and  g(X′, Y, Z) = X′YZ + X′YZ′ + XY
Which one of the following is correct?
(A) Both {f} and {g} are functionally complete
(B) Only {f} is functionally complete
(C) Only {g} is functionally complete
(D) Neither {f} nor {g} is functionally complete

Explanation: A function is considered as functionally complete if it does not belong to T0,T1,L,M,S which are

Property 1: We say that boolean function f preserves zero, if on the 0-input it produces 0. By the 0-input we mean such an input, where every input variable is 0 (this input usually corresponds to the ﬁrst row of the truth table). We denote the class of zero-preserving boolean functions as T0 and write f ∈ T0.

Property 2: Similarly to T0, we say that boolean function f preserves one, if on 1-input, it produces 1. The 1-input is the input where all the input variables are 1 (this input usually corresponds to the last row of the truth table). We denote the class of one-preserving boolean functions as T1 and write f ∈ T1.

Property 3: We say that boolean function f is linear if one of the following two statements holds for f:

• For every 1-value of f, the number of 1’s in the corresponding input is odd, and for every 0-value of f, the number of 1’s in the corresponding input is even.

or

• For every 1-value of f, the number of 1’s in the corresponding input is even, and for every 0-value of f, the number of 1’s in the corresponding input is odd.

If one of these statements holds for f, we say that f is linear1. We denote the class of linear boolean functions with L and write f ∈ L.

Property 4: We say that boolean function f is monotone if for every input, switching any input variable from 0 to 1 can only result in the function’s switching its value from 0 to 1, and never from 1 to 0. We denote the class of monotone boolean functions with M and write f ∈ M.

Property 5: We say that boolean function f(x1,…,xn) is self-dual if f(x1,…,xn) = ¬f(¬x1,…,¬xn).

The function on the right in the equality above (the one with negations) is called the dual of f. We will call the class of self-dual boolean functions S and write f ∈ S.

As in our case we can see  on giving all i/p to 0 (g )produce 0 so it preserving 0 and can’t be functionally complete.

But f is neither preserving 0 nor 1.

• F is not linear(see defn. of linear above)
• F is not monotone(see defn. of monotone above)
• F is not self dual as f(x,y,z) is not equal to –f(-x,-y,-z)

So f is functionally complete.

Hence ans is (B) part

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