The Gateway to Computer Science Excellence
+9 votes

Consider three $4$-variable functions $f_1, f_2$, and $f_3$, which are expressed in sum-of-minterms as



$f_3=\Sigma (2,7,11,14)$

For the following circuit with one AND gate and one XOR gate the output function $f$ can be expressed as:

  1. $\Sigma(7,8,11)$
  2. $\Sigma (2,7,8,11,14)$
  3. $\Sigma (2,14)$
  4. $\Sigma (0,2,3,5,6,7,8,11,14,15)$
in Digital Logic by Veteran (431k points)
edited by | 2.8k views

4 Answers

+16 votes
Best answer
Perform $f_1 \cdot f_2$ first, then with the result perform $XOR$ with $f_3$.

$f1 \cdot f2$ means just take common minterms in $f_1$ and $f_2$ (WHY? due to AND gate present, the minterm should be present in both functions.)

$f_1.f_2 = \Sigma (0,2,5,8,14) \cdot  \Sigma (2,3,6,8,14,15) = \Sigma (2,8,14)$

$\Sigma (2,8,14) \oplus  \Sigma (2,7,11,14) = \Sigma (7,8,11)$
by Veteran (65.6k points)
selected by
∑(2,8,14) ⊕ ∑(2,7,11,14) = ∑(7,8,11)


Sir can you please explain this step?
ex-or means, odd no.of times that minterm exist !

2 appeared even number of times, so it can't be in the result !

8 appeared odd number of times, so it should be in the result !

continue this procedure...

XOR i.e. $\bigoplus$ is related to Symmetric Difference. Let two sets $A=\{2,8,14\}$ and $B=\{2,7,11,14\}$

$A\bigoplus B=A\cup B-A\cap B=\{2,7,8,11,14\}-\{2,14\}=\{7,8,11\}$


@Shaik Masthan

Ex-OR is Odd number of 1's detector i.e. Output is 1, if we have Odd number of input variable as 1

Can you prove that, how it works on Number of minterms?




XOR i.e. ⨁⨁ is related to Symmetric Difference. Let two sets A={2,8,14}A={2,8,14} and B={2,7,11,14}B={2,7,11,14}


This is a good point but i don't think its the fundamental property of $\mathbf{\oplus}$, right?


Nope. It's the fundamental property.
+6 votes
f1 and f2 = ∑(0,2,5,8,14) and ∑(2,3,6,8,14,15) = ∑(2,8,14)  (common minterms)
(f1 and f2) xor f3 = ∑(2,8,14) xor ∑(2,7,11,14)
we know a ⊕ b = a'b + ab', so
∑(2,8,14) xor ∑(2,7,11,14)
= (∑(0,1,3,4,5,6,7,9,10,11,12,13,15) and ∑(2,7,11,14))  or  (∑(2,8,14) and ∑(0,1,3,4,5,6,8,9,10,12,13,15))
= ∑(7,11) or ∑(8)
= ∑(7,8,11)   (union)
by (249 points)
+2 votes

From the  given diagram we can write like

$f=(f_{1}.f_{2})\oplus f_{3}$

We know that $A\oplus B=A\overline{B}+\overline{A}B$

So ,we can expend it and get


Apply De-Morgan's  laws



Now $f=(f_{1}.f_{2}).\overline{f_{3}}+\overline{(f_{1}}+\overline{f_{2})}.f_{3}$

             Apply distributive rule

$f=(f_{1}.f_{2}).\overline{f_{3}}+\overline{f_{1}}.f_{3}+\overline{f_{2}}.f_{3}$  -------------$>(1)$

Given that





$f_3=\Sigma (2,7,11,14)$




Perform Intersection and get it



Perform Intersection and get it


and $\overline{f_{1}}.f_{3}=\Sigma(1,3,4,6,7,9,10,11,12,13,15).\Sigma (2,7,11,14)$

      Perform Intersection and get it


and last one 

$\overline{f_{2}}.f_{3}=\Sigma(0,1,4,5,7,9,10,11,12,13).\Sigma (2,7,11,14)$

 Perform Intersection and get it


Now put the values in the equation $(1)$ and get



Perform Union and get it



This is better way to analyze,how GATE works.

by Veteran (58.9k points)
edited by

$\textbf{Symmetric difference,}$ denoted $\triangle$, refers to the set of elements which are in one of the two sets but not both. For example, $\{1,2,3\} \triangle \{3,4,5\} = \{1,2,4,5\}$. In Boolean logic, symmetric difference is expressed as a logical $\textbf{XOR}$ gate.


0 votes



$(f1.f2)\bigoplus f3$






by Active (4.1k points)
Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
50,737 questions
57,292 answers
104,919 users