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

**Answer:** **(B)** **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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