Gate_2019_Mock_Paper

Question

how to solve dld tautology questions?

Answer 1

Option C is correct.

Tautology means a Valid formula. In digital logic, Any Valid(Always true) boolean expression can thought of as a Tautology. 

In Logic, A propositional logic formula is called Valid/Tautology when it is True under all the possible assignments of  truth values(T/F)  to its Variables. 

for eg : $A + A' = 1$ is a Valid boolean formula as we know. Hence, It is a Valid/Tautology. 

And We know, in digital logic,  if a statement is true, then its dual statement is  also true and this dual statement is obtained by changing $+$ for $.$, and $0$ for $1$ and vice versa.

So, If we take Dual of any Tautology/Valid formula in digital logic then We get another Valid/Tautology formula.

Comments

@gauravkc , could you please tell me ,  Is there any other definition possible for the boolean dual apart from the definition which is given in rosen ?

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I don't understand by what logic this answer is saying truth value of all functions will be same as it's dual. (That is what option C do).

For example f(A,B,C) = 1, how will this value tautology be retained by swapping 0 with 1 then we will have g = 0 . Correct it am wrong.

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@Shubham , you and Soumya both are right. In Rosen , It is mentioned that :-

"An identity between functions represented
by Boolean expressions remains valid when the duals of both sides of the identity are taken.This result, called the duality principle, is useful for obtaining new identities."

For Example :-

Identity 1 :- $(A+B)' = A'.B'$

We can get the new identity by taking dual of both sides :-

So,

Identity 2 :- $ (A.B)' = A' + B' $

I think in the given question , If we replace tautology by identity then given answer will be correct. I was sure about it but I was confirming that is there any other possible  definition of boolean dual or not. According to the definition given in Rosen , given question and answer both are wrong. I am attaching the screenshot from Rosen.

PFA.