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GATE | CS 2013 | Question 59

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Which of the following is/are functionally complete?

(A)

f(a, b, c) = (a + b)(a + c)

(B)

f(a, b, c) = a’b’ + b’c + c’a’

(C)

f(a, b, c) = a’ + bc

(D)

None of the above


Answer: (D)

Explanation:

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

Property 1 (TO): 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 TO and writing f € TO.

Property 2 (T1): Similarly to TO, 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 (L): 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.


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Last Updated : 19 Feb, 2022
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