# Boolean's Algebra - SOP

Discussion in 'Homework Help' started by kelvinmacks, Dec 15, 2014.

1. ### kelvinmacks Thread Starter New Member

Dec 15, 2014
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Can someone please explain why the F( X, Y ) is XY' + XY ? IMO , it is X'Y' + X'Y +XY' +XY .... or the author simplified it to become XY' + XY ? how to simplify it?

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2. ### WBahn Moderator

Mar 31, 2012
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Your four-term expression covers ALL four possibilities, and thus would reduce to F(X,Y)=True.

The expression XY' + XY reduces to just X. Do you see how that is the case?

3. ### kelvinmacks Thread Starter New Member

Dec 15, 2014
19
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no, can you explain further?

Mar 31, 2012
20,075
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5. ### WBahn Moderator

Mar 31, 2012
20,075
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Can you "factor out" the X from both terms?

What does that leave you with?

What is (A+A') equal to?

6. ### kelvinmacks Thread Starter New Member

Dec 15, 2014
19
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here's my working:
(X'Y') + (X'Y) + (XY') + (XY)
=XY +Y'(X' +X ) +X'Y
= XY +Y' +X'Y
=XY +Y' +X'Y
= Y(X +X') +Y'
= Y+Y'

7. ### WBahn Moderator

Mar 31, 2012
20,075
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And what is Y+Y' equal to?

Where did you come up with that first expression? Remember that SOP is the sum of all the minterms for which the output is True. You can't just sum up all of the combinations because then you are saying that the output doesn't matter.

8. ### kelvinmacks Thread Starter New Member

Dec 15, 2014
19
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X(Y+Y') +X'Y +X'Y'
=X +X'Y +X'Y'

Dec 15, 2014
19
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Y+Y' = 1

10. ### WBahn Moderator

Mar 31, 2012
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Okay, so where do these lead you?

11. ### cssc New Member

Oct 19, 2014
26
1
in the attached file,
it is given that
"Here, the SOP is F(X,Y)=XY'+XY"
but I don't think that's true, because,
that equation doesn't satisfy for the values given in the truthtable
like,
X=0,Y=1,F=1
X.Y'+X.Y=(0.0)+(0.1)=0 but not 1

SOP would obviously be
X'Y'+XY'+X'Y+XY=1 (TRUE)

12. ### WBahn Moderator

Mar 31, 2012
20,075
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You are correct that F(X,Y) = XY' + XY does not agree with the table above it, though the table appears to extend to the right of the three columns you have shown, so it could be that there is something there that affects how the equation below it is supposed to be interpreted.

But I don't get how you are saying:

What about X=0, Y=0 or X=1, Y=1? Those are each supposed to yield F=0 but your expression yields F=1 for both (indeed, for ALL combinations).

13. ### kelvinmacks Thread Starter New Member

Dec 15, 2014
19
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1.) according to your notes , A minterm is a Boolean's expression that is true fir the minimum number of combinations of inputs ;this minimum number is exactly one. What do you mean by this? I still cant understand , can you please explain further ?
2.) A maxterm is a Boolean's expression that is true for the maximum number of combinations of inputs; this maximum number is exactly one fewer than the total number of possibilities . What do you mean by this? I still cant understand , can you please explain further ?

14. ### WBahn Moderator

Mar 31, 2012
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Say we have four variables, {A,B,C,D}. If one of our terms is ABC, then this is NOT a minterm because it covers two combinations: It is true if {A,B,C,D} are all True, but is it also True if {A,B,C} are True but {D} is False. In order to be a minterm, it must have ALL four variables (with each variable being either complemented or noncomplemented).

Similarly, A+B+C is False if {A,B,C,D} are all False, but it is also False if {A,B,C} are all False but {D} is True. In order to be a maxterm, it must have ALL four variables (with each variable being either complemented or noncomplemented).

15. ### kelvinmacks Thread Starter New Member

Dec 15, 2014
19
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well , can you explain why the minterm involve multiplication operation , and why the maxterm involve addition operation?

16. ### kelvinmacks Thread Starter New Member

Dec 15, 2014
19
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@WBahn , how do you know to transform A' + B' +CC' into ( A' +B' +C)(A' +B' +C' ) ??

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17. ### WBahn Moderator

Mar 31, 2012
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Well, think about it (and it is described in the blog).

Let's again consider a system with four variables, {A,B,C,D}.

If our goal is to build a minterm and we take a single variable, say A, how many of the possible 16 combinations is it True for? Eight. We want to build an expression that is True for exactly one. We can either AND another variable with it or we can OR another variable with it. Which one will reduce the number of combinations for which the expression is true?

If our goal is to build a maxterm and we take a single variable, say A, how many of the possible 16 combinations is it True for? Eight. We want to build an expression that is True for exactly all BUT one. So we can either AND another variable with it or we can OR another variable with it. Which one will increase the number of combinations for which the expression is true?

18. ### WBahn Moderator

Mar 31, 2012
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Direct application of the distributive property of OR over AND.

19. ### kelvinmacks Thread Starter New Member

Dec 15, 2014
19
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do you mean the F(X, Y) from the book is wrong? so the F(X, Y) should be X'Y'+XY'+X'Y+XY ?

20. ### WBahn Moderator

Mar 31, 2012
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The issue of whether it should be X'Y'+XY'+X'Y+XY has already been addressed. This expression covers ALL FOUR possible combinations and is therefore ALWAYS true. Is the function represented by the truth table always true? If not, then you know that this expression is not the function for that truth table.

What are in the remaining columns in that table? It appears that there are additional columns that are cut off.