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+30 votes
The number of min-terms after minimizing the following Boolean expression is _______.

in Digital Logic by Veteran (105k points)
edited by | 4.8k views
Directly apply the boolean algebra and get the right answer


15 max terms

1 min terms 

11 Answers

+55 votes
Best answer

$F = [D'+AB'+A'C+AC'D+A'C'D]'$

$F'= D'+AB'+A'C+AC'D+A'C'D$

Now we have F', so fill 0's (maxterms) in K-map for each term

As for D'

Similarly for $AB'$, $A'C, AC'D$ and $A'C'D$.  We will get

We get one place for minterm and that is ABCD

by Veteran (57k points)
edited by
@praveen _sir why we need to make k map,

suppose if i put A=B=C=D= 1 and solve the F  then we get 0 so the value of F' = 1

correct sir ??
it will work with this problem
@praveen_sir only this particular problem ??

I always used to solve such type of question , am i going in wrong direction??
actually here we get F= ABCD, on putting 1  , we get  F= 1 , it is only possible with one combination.
if suppose we got F= 0, there are 4 possibilities


Sir m not getting this step



2nd equation above represent in F' function whatever on RHS are minterms (as it is SOP )So ..for K map of F'

we should fill minterms why maxterms filled here..


@jatin khachane 1 

Let $F={[{D}'+A{B}'+{A}'C+A{C}'D+{A}'{C}'D]}'$

Apply both side complement 


${F}'=[{D}'+A{B}'+{A}'C+A{C}'D+{A}'{C}'D]$         $[({X}')'=X]$



${F}'={D}'+A{B}'+{A}'C+1.{C}'D$            $[X+{X}'=1]$


${F}'=({D}'+{C}').({D}'+D)+A{B}'+{A}'C$          $[X+YZ=(X+Y).(X+Z)]$












Apply both side complement


Apply Demorgan's laws





So,this is the minterm.

I understood this method ..but not getting that K map method :(
@jatin we've complimented the function now where there was min terms they got changed to max terms and maxterms to minterms in K-map.

In K-map method D' means we have a subcube of size 8 beacuse of which three variables got eliminated we form a Subcube of size 8 in K Map case of AC' ..two variables are missing therefore we've to form a subcube of size 4 in the K-map by filling the max terms wise we've to fill for all.

So is it like this

# of minterms in F = ? 

We know # of minterms in F = # of maxterms in F'

F = [D′+AB′+A′C+AC′D+A′C′D]′

So, F′ = [D′+AB′+A′C+AC′D+A′C′D]

K map for F'

  C'D' C'D CD CD'
A'B' 1 1 1 1
A'B 1 1 1 1
AB 1 1 0 1
AB' 1 1 1 1

Maxterm in F' => A'+B'+C'+D'

So in F only one Minterm ABCD 

Correct me if i am wrong?

Yes it is right

and you also see my answer, below the comment section.
Yes ..i have seen ..that will be very easy..just to make concept clear on mint and max t..i did this ..
Good :)
+1 is always true that
1) # of minterms in F = # maxterms in its complemented form.
2) So Demorgans law is the only way to find complement of the function

@jatin khachane 1 Please don't say sir to me

I'm also an aspirant like yours.

I think both statements are looking right.


@jatin khachane 1 don't get confused unnecessarily. Stick to basics. Recall basic functions that we study in discrete.

1. If there are total n terms in domain and x of which are mapped to 1 then n-x will be mapped to 0.

2. f(n)=1 all combination of input where f gives 1. Complement of f : remaining combination where f(n) gives 0.

3. Check question we are asked about combination where f(n) = 1. But function is complimented that means bracket will give all the terms which will be mapped to 0.(all terms except 15)

4. If you take compliment the result is combination which is mapped to 1.

5. Hence the canonical collection inside bracket represents max terms (mapping to 0) hence sir has written 0 in k map.

There might be a chance that it a question could be proven wrong with other outputs :)
+51 votes

Let's First Simplify it 

[D' + AB' + A'C + AC'D + A'C'D ]'

[D' + AB' + A'C + C'D (A + A') ]'     // A + A' =1

[AB' +A'C + (D' + C') (D' + D) ]'    // Apply Distributive Rule Among D' and C'D

[AB' + A'C + D' + C' ]'

[AB' + (A' + C') (C + C') + D'  ] '   //Apply distributive Law B/w A'C and C'

[AB' + A' +C' +D' ]'

[(A+ A') (A' + B') +  C' + D' ]'    //Apply Distributive law b/w AB' and A'

Finally we got

[A' +B' +C' + D' ]'

Apply Demorgan's Law


by Boss (45.4k points)
+15 votes
Canonical SOP of the expression D'+AB'+A'C+AC'D+A'C'D is ∑m(0,1,2,3,4,5,6,7,9,13,12,14,11,8,10)

So corresponding POS form is ⫪M(15)



Therefore number of minterms = 1
by Junior (539 points)
+7 votes
answer is 1

solve it we get 15 max terms

so no. of min terms are 1
by Active (1.4k points)
+7 votes












Remove similar tearm,because $[X+X=X]$


Apply both side complement


Apply Demorgan's laws



$F=(A'+B'+C'+D).(A'+B'+C+D).(A'+B+C'+D).(A'+B+C+D).(A+B'+C'+D).(A+B'+C+D).(A+B+C'+D).(A+B+C+D).(A'+B+C'+D').(A'+B+C+D').(A+B'+C'+D').(A+B+C'+D').(A'+B'+C+D').(A+B'+C+D').(A+B+C+D')$                 $[(X')'=X]$

This is Canonical Product Of Sum Term(Maxterm)
















$F(A,B,C,D)=\prod (1,2,3,4,5,6,7,8,9,,10,11,12,13,14)$

$F(A,B,C,D)=\sum (15)$

      $$ \textbf{(OR)}$$

Let's First Simplify it



$F=[D'+AB'+A'C+C'D.1]'$        $[A+A'=1]$


$F=(D')'.(AB')'.(A'C)'.(C'D)'$       [ Using Demorgan's  Law$: (A+B)=A'.B'$  $(or)$ $(A.B)'=A'+B' ]$

$F=(D).(A'+B).(A+C').(C+D')$    [Again using Demorgan's law ]

$F=(A'D+BD).(AC+AD'+C'C+C'D')$   [Simple multiply]

$F=(A'D+BD).(AC+AD'+0+C'D')$     $[C.C'=0]$



$F=ABCD$       $[A.A'=0,D.D'=0]$


Let $f(A,B,C,D) = \bigg[D'+AB'+A'C+AC'D+A'C'D\bigg]'$

$\implies \bigg[D'+AB'+A'C+C'D\bigg]'$

$\implies \bigg[D'+C'+ AB' + A'C\bigg]'$

$\implies \bigg[D'+C'+A'+ AB'\bigg]'$

$\implies \bigg[D'+C'+A'+ B'\bigg]'$​​​​​​​

$\implies ABCD$​​​​​​​

So the number of min-terms$=1$

by Veteran (58.8k points)
edited by

jatin khachane 1 

see my answer


BTW in the part of your simplification, it can be done in shorter way like this below (especially in the part of distribution).

$\begin{align} F &=\overline{\overline{D}+A\overline{B}+\overline{A}C+A\overline{C}D+\overline{A}\overline{C}D}\\&=\overline{\overline{D}+A\overline{B}+\overline{A}C+\overline{C}D};~~[\because (A+\overline{A})\overline{C}D=1\cdot \overline{C}D]\\&=D(\overline{A}+B)(A+\overline{C})(C+\overline{D});~~[\text{Applying De Morgan's law}]\\&=(\overline{A}+B)(A+\overline{C})(CD+\overline{D}D)\\&=(0+\overline{A}\overline{C}+AB+B\overline{C})(CD+0);~~[\because X\cdot\overline{X}=0]\\&=(\overline{A}\overline{C}+AB+B\overline{C})CD\\&=0+ABCD+0\\&=ABCD \end{align}$

+6 votes
on putting some element say A = 0 we solve the boolean expression we get ans = 0; so for all minterms where A=0 function is 0. On taking A=1 we solve the boolean expression and get function = BCD, on K-Map its easy to put value of BCD.

The end result of this gives us only one minterm = ABCD

hence, answer = 1
by Boss (30.8k points)
+2 votes
Answer: 1
by Boss (33.9k points)
+2 votes
The minterm we get after minimizing this expression is ABCD.
So, number of minterms is 1.
by Junior (939 points)

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