GATE CSE 2024 | Set 2 | Question: 40

Question

Consider $4$-variable functions $f 1, f 2, f 3, f 4$ expressed in sum-of-minterms form as given below.
\[
\begin{array}{l}
f 1=\sum(0,2,3,5,7,8,11,13) \\
f 2=\sum(1,3,5,7,11,13,15) \\
f 3=\sum(0,1,4,11) \\
f 4=\sum(0,2,6,13)
\end{array}
\]

With respect to the circuit given above, which of the following options is/are CORRECT?

• A. $\boldsymbol{Y}=\sum(0,1,2,11,13)$
• B. $\boldsymbol{Y}=\Pi(3,4,5,6,7,8,9,10,12,14,15)$
• C. $\boldsymbol{Y}=\sum(0,1,2,3,4,5,6,7)$
• D. $\boldsymbol{Y}=\Pi(8,9,10,11,12,13,14,15)$

Answer 1

Here AND gate is performed as intersection operations as it will return common elements between $f_1,f_2$

OR gate is performed as a union operation as it will return all the elements present in both $f_3,f_4$

Assume the output of the AND gate is $y_1$ and the output of the OR gate is $y_2$.

$y_1=f_1.f_2=\sum_m(3,5,7,11,13)$

$y_2=f3+f_4=\sum_m(0,1,2,4,6,11,13)$

Now both $y_1,y_2$ act as input for XOR gate which returns the elements that is in either in $y_1$ or in $y_2$. It will discard the common elements between $y_1,y_2$.

So the output $Y=y_1\oplus y_2=\sum_m(0,1,2,3,4,5,6,7) \equiv \Pi_m(8,9,10,11,12,13,14,15)$

So Option $(C, D)$ is correct.

A similar type of concept asked in:

1. GATE CSE 2002 | Question: 2-1
2. GATE CSE 2008 | Question: 8
3. GATE CSE 2020 | Question: 28

Comments

What if instead of XOR ,XNOR was given before Y?

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@Silver_Reaper then it will give those minterms which are common in both as well as absent in both.

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@AniMan_7 Do you mean to say that minterms which are jointly missing from both expressions,only those will be part of the result?

Also can you refer me to a source for this?

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yes, jointly missing and jointly present also. cz $A \odot B = AB + \overline{A} \ \overline{B}$
Source: https://deepai.org/machine-learning-glossary-and-terms/xnor