I usually try not ask for homework help but once again I am stuck. I've been going over and over my textbook but I am not able to figure this out. Emailed the instructor and all the help i get is "Check this page" and "check that page", so instead of just not doing it, I would like some advice so I am actually able to learn.
The "G" gate thing, is whats bugging me in the book there is no gate that looks like that so i have no idea what to do. Here's a picture of the question, basically I have to find the X values (outputs). The answer would be nice. But I highly would appreciate a little explaination of "why".
Really appreciate the help!
Ok, so you you're not quite sure what this G gate means from looking at the diagram. In a question like this (where there's something that doesn't make sense to you), it's helpful to start with what you do know.
From looking at the diagram I don't know what the G gate means. In fact I don't know anything about circuits (but I do know something about logic :) ). I start with the truth table that the author has generously given me the formulas for each gate. I notice that there are 3 operators (*,+,') which I know.
If you know what those operators mean, then you can derive the meaning of the G gate.
D looks like an AND
E looks like NAND, the nipple-ish thing is an inverter
F looks like NOT, a buffer with an inverter on the output
G looks like a NOR, an OR with inverted output
The unlabeled one looks like an OR.
That's a really bad drawing though.
Giving you the answer would only cheat you out of your education and this stuff is important. There are 16 logical connectives for binary functions and they're all . . . logical. They make sense.
AND means when both inputs are true the output will be true. "If A and B = 1 output is 1"
OR means if any of the inputs are true the output will be true. "if A or B = 1 output is 1"
NOT means if the input is true the output is false.
XOR means if either input is true, but not both, the output will be true. "If A or B = 1 output is 1 unless both A and B = 1"
AND, OR, and XOR can all have inverters on their outputs which reverses their meanings. When they're supposed to output true they'll output false and when they're supposed to output false they'll output true.
The headings in the table are using * to mean AND, + to mean OR, and ' to mean "invert the symbol on the left".
D is A AND B, so if A and B are true, then put a 1 in the column, the rest of the column is false.
E is B NAND C, so if B and C are true, then put a 0 in the column, the rest of the column is true.
F is NOT C, so put the opposite of C in the column.
G is NOT((A AND B) OR (NOT C)), or if you look at the schematic and think about the formulas a bit you'll see that it's NOT(D OR F). You should be able to figure this out on your own now.
X is G OR E, There's a more complicated formula for it that traces through the circuit like the formula for G, but if you need it to prove your work you'll have to talk to your teacher. You'll probably get more help asking questions that show you put in effort.
I wrote an article about Logical functions in JavaScript that includes schematics. If you memorize the function tables at the top of the article it'll help you a lot when dealing with digital logic. Bonus points for associating the function number with the function name, you'll have memorized the output column of the functions truth table. They've taught you to count in binary right? Anyway, here's the article: http://matthewkastor.blogspot.com/2013/10/logical-functions-in-javascript.html It's not so important for your immediate question but will definitely do you good to read it. Oh, inputs can be inverted as well so don't let that throw you off.
Related
I have a problem with some homework, for my Advanced Digital Design course, in which I have to create the truth table and find the Boolean expression for a provided circuit (image is linked below). I was able to create the truth table and I think find the Boolean Expression for the problem using the truth table that I created, but I don't think that this is the way that we are supposed to find the Boolean Expression. I was hoping that someone could share some insight on how to find the Boolean Expression without using the truth table.
I would normally not have such an issue with this, but since there are 5 variables, and NAND gates, I am quite confused on how to simplify once I find it.
I think that the outcome is something like:
[(a(bcd)'e)']' when you look at the circuit, and not the table, but I am not entirely sure. I am also not sure on how to simplify this into a Boolean expression if this is right.
Using the truth table to find the minterms I get y= m17+m19+m21+m22+m23+m25+m29 (which I am also not sure if it is right). And if I use a K-Chart to solve this, I end up with y = ab'e + a'ce + ab'cd, which seems like a legitimate simplified Boolean expression, but I have no clue if that is right.
Since this question is worth 20 of the total 100 points, I could really use some help understanding how this works.
Here is the image we were provided:
Circuit: only circuit (a), not (b)
Thank you!
I think that the outcome is something like: [(a(bcd)'e)']'
Your first guess is correct.
You just have to remark that whatever''=whatever
f=[(a(bcd)'e)']' = a(bcd)'e
Using de Morgan (bcd)' = b'+c'+d'
Hence f=a(bcd)'e = ae(b'+c'+d') =ab'e + ac'e + ad'e
which is minimal.
This has been bugging me since a long time.
Suppose I have a boolean function F defined as follows:
Now, it can be expressed in its SOP form as:
F = bar(X)Ybar(Z)+ XYZ
But I fail to understand why we always complement the 0s to express them as 1. Is it assumed that the inputs X, Y and Z will always be 1?
What is the practical application of that? All the youtube videos I watched on this topic, how to express a function in SOP form or as sum of minterms but none of them explained why we need this thing? Why do we need minterms in the first place?
As of now, I believe that we design circuits to yield and take only 1 and that's where minterms come in handy. But I couldn't get any confirmation of this thing anywhere so I am not sure I am right.
Maxterms are even more confusing. Do we design circuits that would yield and take only 0s? Is that the purpose of maxterms?
Why do we need minterms in the first place?
We do not need minterms, we need a way to solve a logic design problem, i.e. given a truth table, find a logic circuit able to reproduce this truth table.
Obviously, this requires a methodology. Minterm and sum-of-products is mean to realize that. Maxterms and product-of-sums is another one. In either case, you get an algebraic representation of your truth table and you can either implement it directly or try to apply standard theorems of boolean algebra to find an equivalent, but simpler, representation.
But these are not the only tools. For instance, with Karnaugh maps, you rewrite your truth table with some rules and you can simultaneously find an algebraic representation and reduce its complexity, and it does not consider minterms. Its main drawback is that it becomes unworkable if the number of inputs rises and it cannot be considered as a general way to solve the problem of logic design.
It happens that minterms (or maxterms) do not have this drawback, and can be used to solve any problem. We get a trut table and we can directly convert it in an equation with ands, ors and nots. Indeed minterms are somehow simpler to human beings than maxterms, but it is just a matter of taste or of a reduced number of parenthesis, they are actually equivalent.
But I fail to understand why we always complement the 0s to express them as 1. Is it assumed that the inputs X, Y and Z will always be 1?
Assume that we have a truth table, with only a given output at 1. For instance, as line 3 of your table. It means that when x=0, y=1 and z=0 , the output will be zero. So, can I express that in boolean logic? With the SOP methodology, we say that we want a solution for this problem that is an "and" of entries or of their complement. And obviously the solution is "x must be false and y must be true and z must be false" or "(not x) must be true and y must be true and (not z) must be true", hence the minterm /x.y./z. So complementing when we have a 0 and leaving unchanged when we have a 1 is way to find the equation that will be true when xyz=010
If I have another table with only one output at 1 (for instance line 8 of your table), we can find similarly that I can implement this TT with x.y.z.
Now if I have a TT with 2 lines at 1, one can use the property of OR gates and do the OR of the previous circuits. when the output of the first one is 1, it will force this behavior and ditto for the second. And we directly get the solution for your table /xy/z+xyz
This can be extended to any number of ones in the TT and gives a systematic way to find an equation equivalent to a truth table.
So just think of minterms and maxterms as a tool to translate a TT into equations. What is important is the truth table (that describes the behaviour of what you want to do) and the equations (that give you a way to realize it).
I have been struggling with some questions from my study guide and really am stuck - I have asked the lecturer for help but his answer was literally "but it's been done for you" (referring to gauss_seidel code that was written) - to which I think he missed the point. I'm struggling to understand the actual question and how to approach it.
The first question reads as follows:
Define the 100x100 square matrix A and the column vector b by:
A(ij)=I(ij)+1/((i-j)2+1) b_(i)=1+2/i 1<=i j<=100
where I_(ij) is the 100x100 identity matrix (i.e 1 on the main diagonal and 0 everywhere else). Solve for x using both the Gauss-Seidel method and the A\b construct.
We have written the code for the gauss_seidel method, and i think i understand what it does mostly, however, i do not understand how the above question fits into the method. I was thinking that i'm supposed to do something like the following in the octave window then calling the gauss_seidel method:
>> A=eye(100,100);
>> b= (this is where i get slightly confused)... I've tried doing
>> for b=1:n;
>> b=1+(2/n);
That is question 1.
Question 2 I have given an answer and asked him about but he has not responded.
It reads: The Hilbert matrix is a square n x n matrix defined by:
H_(ij)n = 1/i+j+1
Define bn to be a column vector of dimension n, and with each element 1. Construct bn and then solve for x, Hn xn=bn in the cases n=4.
What i did here was simply:
>> b=ones (4,1);
>> x=hilb(4)\b;
and then it gave me the output of x values. Im not sure if what i did here was correct... since it doesnt mention using any method at all it just says solve for x.
Im not sure how to relate the lecturers reply to understanding the problem.
If you could help me by maybe letting me know what im missing or how i should be thinking about this, it would really help.
the gauss_seidel code looks like this:
function xnew=gauss_seidel(A,b,xold)
n=size(A)(1);
At=A;
xnew=xold;
for k=1:n
At(k,k)=0;
end
for k=1:n
xnew(k)=(b(k)-At(k,:)*xnew)/A(k,k);
end
endfunction
Ive been writing pseudo since last Monday and I am only a little bit clearer on what the code does.
A(ij)=I(ij)+1/((i-j)2+1), b(i)=1+2/i, 1<=i, j<=100
All this is really saying is that we have to create A and b in such a way that i>=1 and j<=100. After doing that, you simply solve using the Gauss Seidel method.
So we'd create b like this:
b=zeros(100,1);
for k=1:100
b(k) = 1+(2/k);
end
This will create a column vector with a size of 100x1 with all the values that satisfy b(i)=1+2/i where i (or in the code,'k') was greater or equal to 1.
Then to create A :
myMatrix=zeros(100,100);
for i=1:100
for j=1:100
myMatrix(i,j) = 1/(((i-j)^2) + 1);
end
end
A=eye(100) + myMatrix;
Now we have created A in such a way that it equals A(ij)=I(ij)+1/((i-j)2+1) where i was greater or equal to 1 & j was less than or equal to 100.
The rest of the question is basically asking to to solve for the values of x using the Gauss Seidel method.
So it be something like this :
y=iterative_linear_solve(A,b,x0,TOL,max_it,method);
Don't forget about creating x0 as the initial assumption, tolerance and max iterations etc.
In terms of question 2, you did exactly what I would have done. I think you're good with that.
I'm not too sure how to answer this :
If you could help me by maybe letting me know what im missing or how i
should be thinking about this, it would really help.
All I can really say is that you need to look at the problems in such a way that you see Ax=b. For example in the first question we started by making b, and then A. After that we simply applied the A\b construct or the Gauss Seidel method and got our answer.
And that's essentially what you did for the second question.
Lastly, are you a UNISA student by chance? I am, haha. I've been struggling with this on my own for a while. The study guides don't seem to give a lot of info.
I've got to the section on operators in The Ruby Programming Language, and it's made me think about operator associativity. This isn't a Ruby question by the way - it applies to all languages.
I know that operators have to associate one way or the other, and I can see why in some cases one way would be preferable to the other, but I'm struggling to see the bigger picture. Are there some criteria that language designers use to decide what should be left-to-right and what should be right-to-left? Are there some cases where it "just makes sense" for it to be one way over the others, and other cases where it's just an arbitrary decision? Or is there some grand design behind all of this?
Typically it's so the syntax is "natural":
Consider x - y + z. You want that to be left-to-right, so that you get (x - y) + z rather than x - (y + z).
Consider a = b = c. You want that to be right-to-left, so that you get a = (b = c), rather than (a = b) = c.
I can't think of an example of where the choice appears to have been made "arbitrarily".
Disclaimer: I don't know Ruby, so my examples above are based on C syntax. But I'm sure the same principles apply in Ruby.
Imagine to write everything with brackets for a century or two.
You will have the experience about which operator will most likely bind its values together first, and which operator last.
If you can define the associativity of those operators, then you want to define it in a way to minimize the brackets while writing the formulas in easy-to-read terms. I.e. (*) before (+), and (-) should be left-associative.
By the way, Left/Right-Associative means the same as Left/Right-Recursive. The word associative is the mathematical perspective, recursive the algorihmic. (see "end-recursive", and look at where you write the most brackets.)
Most of operator associativities in comp sci is nicked directly from maths. Specifically symbolic logic and algebra.
I need some help with this Boolean Implication.
Can someone explain how this works in simple terms:
A implies B = B + A' (if A then B). Also equivalent to A >= B
Boolean implication A implies B simply means "if A is true, then B must be true". This implies (pun intended) that if A isn't true, then B can be anything. Thus:
False implies False -> True
False implies True -> True
True implies False -> False
True implies True -> True
This can also be read as (not A) or B - i.e. "either A is false, or B must be true".
Here's how I think about it:
if(A)
return B;
else
return True;
if A is true, then b is relevant and should be checked, otherwise, ignore B and return true.
I think I see where Serge is coming from, and I'll try to explain the difference. This is too long for a comment, so I'll post it as an answer.
Serge seems to be approaching this from the perspective of questioning whether or not the implication applies. This is somewhat like a scientist trying to determine the relationship between two events. Consider the following story:
A scientist visits four different countries on four different days. In each country she wants to determine if rain implies that people will use umbrellas. She generates the following truth table:
Did it rain? Did people Does rain => umbrellas? Comment
use umbrellas?
No No ?? It didn't rain, so I didn't get to observe
No Yes ?? People were shielding themselves from the hot sun; I don't know what they would do in the rain
Yes No No Perhaps the local government banned umbrellas and nobody can use them. There is definitely no implication here.
Yes Yes ?? Perhaps these people use umbrellas no matter what weather it is
In the above, the scientist doesn't know the relationship between rain and umbrellas and she is trying to determine what it is. Only on one of the days in one of the countries can she definitively say that implies is not the correct relationship.
Similarly, it seems that Serge is trying to test whether A=>B, and is only able to determine it in one case.
However, when we are evaluating boolean logic we know the relationship ahead of time, and want to test whether the relationship was adhered to. Another story:
A mother tells her son, "If you get dirty, take a bath" (dirty=>bath). On four separate days, when the mother comes home from work, she checks to see if the rule was followed. She generates the following truth table:
Get dirty? Take a bath? Follow rule? Comment
No No Yes Son didn't get dirty, so didn't need to take a bath. Give him a cookie.
No Yes Yes Son didn't need to take a bath, but wanted to anyway. Extra clean! Give him a cookie.
Yes No No Son didn't follow the rule. No cookie and no TV tonight.
Yes Yes Yes He took a bath to clean up after getting dirty. Give him a cookie.
The mother has set the rule ahead of time. She knows what the relationship between dirt and baths are, and she wants to make sure that the rule is followed.
When we work with boolean logic, we are like the mother: we know the operators ahead of time, and we want to work with the statement in that form. Perhaps we want to transform the statement into a different form (as was the original question, he or she wanted to know if two statements are equivalent). In computer programming we often want to plug a set of variables into the statement and see if the entire statement evaluates to true or false.
It's not a matter of knowing whether implies applies - it wouldn't have been written there if it shouldn't be. Truth tables are not about determining whether a rule applies, they are about determining whether a rule was adhered to.
I like to use the example: If it is raining, then it is cloudy.
Raining => Cloudy
Contrary to what many beginners might think, this in no way suggests that rain causes cloudiness, or that cloudiness causes rain. (EDIT: It means only that, at the moment, it is not both raining and not cloudy. See my recent blog posting on material implication here. There I develop, among other things, a rationale for the usual "definition" for material implication. The reader will require some familiarity with basic methods of proof, e.g. direct proof and proof by contradiction.)
~[Raining & ~Cloudy]
Judging from the truth tables, it is possible to infer the value of a=>b only for a=1 and b=0. In this case the value of a=>b is 0. For the rest of values (a,b), the value of a=>b is undefined: both (a=>b)=0 ("a doesn't imply b") and (a=>b)=1 ("a implies b") are possible:
a b a=>b comment
0 0 ? it is not possible to infer whether a implies b because a=0
0 1 ? --"--
1 0 0 b is 0 when a is 1, so it is possible to conclude
that a does not imply b
1 1 ? whether a implies b is undefined because it is not known
whether b can be 0 when a=1 .
For a to imply b it is necessary and sufficient that b=1 always when a=1, so that there is no counterexample when a=1 and b=0. For the rows 1, 2 and 4 in the truth table it is not known whether there is counterexample: these rows do not contradict to (a=>b)=1, but they also do not prove (a=>b)=1 . In contrast, row 3 immediately disproves (a=>b)=1 because it provides a counterexample when a=1 and b=0.
I guess I may shock some readers with these explanations, but it seems there are severe errors somewhere in the basics of the logic we are taught, and that is one of the reasons for such problems as Boolean Satisfiability being not solved yet.
The best contribution on this question is given by Serge Rogatch.
Boolean logic applies only where the result of quantifying(or evaluation) is either true or false and the relationship between boolean logic propositions is based on this fact.
So there must exist a relationship or connection between the propositions.
In higher order logic, the relationship is not just a case of on/off, 1/0 or +voltage/-voltage, the evaluation of a worded proposition is more complex. If no relationship exists between the worded propositions, then implication for worded propositions is not equivalent to boolean logic propositions.
While the implication truth table always yields correct results for binary propositions, this is not the case with worded propositions which may not be related in any way at all.
~A V B truth table:
A B Result/Evaluation
1 1 1
1 0 0
0 1 1
0 0 1
Worded proposition A: The moon is made of sour cream.
Worded proposition B: Tomorrow I will win the lotto.
A B Result/Evaluation
1 ? ?
As you can see, in this case, you can't even determine the state of B which will decide the result. Does this make sense now?
In this truth table, proposition ~A always evaluates to 1, therefore, the last two rows don't apply. However, the last two rows always apply in boolean logic.
http://thenewcalculus.weebly.com
Here's a compact statement:
Suppose we have two statements, A and B, each of which could either be true or false. Without any further information, there are 2 x 2 = 4 possibilities: "A and not B", "B and not A", "neither A nor B", and "both A and B".
Now impose the additional restriction that "if A, then also B". After imposing this restriction, the expression "x -> y", where -> is the "implication" operator, denotes whether it is still possible for A == x and B == y. The only outcome that is no longer possible after this additional restriction is A == 1 and B == 0, since that contradicts the restriction itself. Hence, we have 1 -> 0 is zero, and every other pair is 1.