GATE | CS 2013 | Question 59
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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