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).

i am sorry, i was wrong
SOP will not be XY+X'Y+XY'+X'Y'

and
SOP would actually be
F(X,Y)=XY'+X'Y (as it is clear that it is an xor truthtable)

 

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.

can you suggest and link or book for me to facilitate my understanding on SOP and POS ? i am confused now...

 

i am sorry, i was wrong
SOP will not be XY+X'Y+XY'+X'Y'

and
SOP would actually be
F(X,Y)=XY'+X'Y (as it is clear that it is an xor truthtable)

why?

 

it is clear that it is an exclusive-or truthtable(an inequality detector)
and the equation of exclusive-or of A and B is
AB'+A'B
so, F(X,Y)=XY'+X'Y

I hope you are aware of exclusive-or operation...

OR ELSE
just take the help of k-maps for finding SOP and POS of any function from its truthtable
and obviously you will end up with the same answer

 

why?

Look at the original truth table. You want the function F to be TRUE if either (X is False AND Y is True) OR (X is True and Y is False).

Now just express that using Boolean algebra notation:

F(X,Y) = (X AND Y') OR (X' AND Y) = XY' + X'Y

 

Look at the original truth table. You want the function F to be TRUE if either (X is False AND Y is True) OR (X is True and Y is False).

Now just express that using Boolean algebra notation:

F(X,Y) = (X AND Y') OR (X' AND Y) = XY' + X'Y

yes , minterm means the function is true , so why the F(X, Y) isn't done by adding all the minterm? but only the last minterm?

say the ans , F(X , Y) = XY' + X'Y
= (1)(1) +(1)(1)
= 1
but if F(X , Y) =X'Y' + X'Y +XY' + X'Y
= (1)(1) +(1)(1) +(1)(1) +(1)(1)
= 1

 

as we already know exclusive-or truthtable., its being easy for us to find sop for this truhtable...
but if we are unable to find it from truthtable.,
use k-maps for finding minimal sop or pos

 

yes , minterm means the function is true , so why the F(X, Y) isn't done by adding all the minterm? but only the last minterm?

say the ans , F(X , Y) = XY' + X'Y
= (1)(1) +(1)(1)
= 1
but if F(X , Y) =X'Y' + X'Y +XY' + X'Y
= (1)(1) +(1)(1) +(1)(1) +(1)(1)
= 1

How can X and X' both be True simultaneously?

There are four possible combination:

F(X,Y) = XY'+X'Y)
F(0,0) = (0)(0')+(0')(0) = (0)(1)+(1)(0) = 0+0 = 0
F(0,1) = (0)(1')+(0')(1) = (0)(0)+(1)(1) = 0+1 = 1
F(1,0) = (1)(0')+(1')(0) = (1)(1)+(0)(0) = 1+0 = 1
F(1,1) = (1)(1')+(1')(1) = (1)(0)+(0)(1) = 0+0 = 0

 

How can X and X' both be True simultaneously?

There are four possible combination:

F(X,Y) = XY'+X'Y)
F(0,0) = (0)(0')+(0')(0) = (0)(1)+(1)(0) = 0+0 = 0
F(0,1) = (0)(1')+(0')(1) = (0)(0)+(1)(1) = 0+1 = 1
F(1,0) = (1)(0')+(1')(0) = (1)(1)+(0)(0) = 1+0 = 1
F(1,1) = (1)(1')+(1')(1) = (1)(0)+(0)(1) = 0+0 = 0

do u mean the X'Y' is incorrect because X and Y cant be false simultaneously? so the F(X, Y ) should be = X'Y +XY' + X'Y , am i right?

 

do u mean the X'Y' is incorrect because X and Y cant be false simultaneously? so the F(X, Y ) should be = X'Y +XY' + X'Y , am i right?

You said the following in the post I was responding to:

but if F(X , Y) =X'Y' + X'Y +XY' + X'Y
= (1)(1) +(1)(1) +(1)(1) +(1)(1)
= 1

In that second line, what is X equal to? What is X' equal to? You are claiming that X, X', Y, and Y' are ALL equal to 1.

X and Y are two signals. At any given moment in time, each signal takes on one of two values, True or False. The two signals are independent, meaning that knowing the value of one of them doesn't tell you anything about the value of the other.

So X and Y can take on any of the four possible combinations, including both being False.

But the function that is defined is True only if one of the signals is True and the other if False. It is supposed to be False if either both signals are True or Both signals are False.

Perhaps an word example might help. I tell my daughter that she must either go to a museum or go to the zoo this weekend. She must do one of them because I want to her learn something this weekend, but she cannot do both because I can't afford that. She can indicate that she is going to go to the museum by flipping switch X to True (False if she isn't going to go to the museum). Similarly, she can indicate that she is going to go to the zoo with switch Y. I build a circuit, F, that lights a light (i.e., is True) if she has made acceptable choices for X and Y (the museum and the zoo).

So she can certainly set the switches to indicate that she isn't going to the museum (X=False) and that she isn't going to the zoo (Y=False), meaning that X' is True and Y' is True, and so X'Y' is True. But this is not a valid option for her and so the function, F, should be False whenever she makes this choice.

But while she can make invalid choices, she can't set one of the switches, say X, to be both True and False at the same time. What does it mean for her to choose to go to the museum but also choose to not go to the museum? Either X is True or X is False, it can't be both. Hence the term (XX') is always False because it is either (0)(1) or it is (1)(0), both of which are False. Conversely, the term (X+X') is always True because it is either (0+1) or it is (1+0), both of which are True.

 

You said the following in the post I was responding to:


In that second line, what is X equal to? What is X' equal to? You are claiming that X, X', Y, and Y' are ALL equal to 1.

X and Y are two signals. At any given moment in time, each signal takes on one of two values, True or False. The two signals are independent, meaning that knowing the value of one of them doesn't tell you anything about the value of the other.

So X and Y can take on any of the four possible combinations, including both being False.

But the function that is defined is True only if one of the signals is True and the other if False. It is supposed to be False if either both signals are True or Both signals are False.

Perhaps an word example might help. I tell my daughter that she must either go to a museum or go to the zoo this weekend. She must do one of them because I want to her learn something this weekend, but she cannot do both because I can't afford that. She can indicate that she is going to go to the museum by flipping switch X to True (False if she isn't going to go to the museum). Similarly, she can indicate that she is going to go to the zoo with switch Y. I build a circuit, F, that lights a light (i.e., is True) if she has made acceptable choices for X and Y (the museum and the zoo).

So she can certainly set the switches to indicate that she isn't going to the museum (X=False) and that she isn't going to the zoo (Y=False), meaning that X' is True and Y' is True, and so X'Y' is True. But this is not a valid option for her and so the function, F, should be False whenever she makes this choice.

But while she can make invalid choices, she can't set one of the switches, say X, to be both True and False at the same time. What does it mean for her to choose to go to the museum but also choose to not go to the museum? Either X is True or X is False, it can't be both. Hence the term (XX') is always False because it is either (0)(1) or it is (1)(0), both of which are False. Conversely, the term (X+X') is always True because it is either (0+1) or it is (1+0), both of which are True.

thanks for the in-depth explaination! so the F(X , Y) = XY' + X'Y??? but not F(X , Y) = XY' + XY... OH! No, the book is giving wrong information!

 

thanks for the in-depth explaination! so the F(X , Y) = XY' + X'Y??? but not F(X , Y) = XY' + XY... OH! No, the book is giving wrong information!

Books will tend to do that. For several reasons. The author may have messed up -- perhaps they changed the problem but not the solution, perhaps they just had a brain fart, perhaps they just made a silly typo. The editor/publisher may have screwed up. The grad student that they paid slave wages to to work all the problems and prepare the solutions manual may have screwed up. Whenever you don't agree with a provided answer, take it as an opportunity to solve a real problem, namely whose solution is actually correct. Don't blindly assume that the book is right, but don't assume that it is wrong too quickly, either. When there is disagreement, the goal is to PROVE which one is wrong (and perhaps both are) and be able to defend your conclusions -- you will learn a LOT in the process.

 

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?

As Wikipedia says, there are multiple issues here.
- Your book has a misprint. The 'X' in the second product-term should be complemented.
- The phrase 'sum of products' means 'sum of TRUE products': only two of the terms are true and are to be included in the sum