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Questions tagged [boolean-algebra]

In mathematics and mathematical logic, boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. Boolean algebra is used in the analysis and simplification of digital (logic) circuits

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When should we NOT simplify a digital signal in an expression?

When we do a simplification in a Boolean expression, sometimes we eradicate some Boolean variables, either from part of the expression or entirely if it's not needed. Is there a case in digital design ...
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Minterms from Logic Circuit

I have the following logic circuit: And I'm asked to identify the minterms in F's truth table. The expression for F is F = C + A'B' and the truth table isn't too difficult to construct (X=F): I ...
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Finding a minimal cost function from a Karnaugh Map

Below is a problem I made up and did. Is my solution wrong? Problem: Attached is following map for the function \$f\$. We want a minimal cost expression using only and, or and not gates. The cost ...
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1answer
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Implement boolean expression with AOI

Below is a question in my homework: \$F(P,Q,R,S,T)=(P+Q)S+(R+T)\bar S\$ using one or more 2x2 AOI."> Here's my attempt for (a): Step 1: simplify boolean expression. \$F(P,Q,R,S,T)\$ \$=(...
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Tricky problem regarding CMOS

My lecturer gave us the following problem today: "A CMOS gate (with inputs A, B, C) consists of a pull-up network with 0 or more PMOS transistors, and a pull-down network with 0 or more NMOS ...
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Proving something is true in Boolean Algebra given two equations and two variables without using a Truth Table

Problem: Given that $$ AB = 0$$ and $$A + B = 1$$ and using only algebraic manipulation is it possible to prove that: $$ \overline A = B$$ Answer: I believe it is, but I have failed in my attempt ...
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Simplifying a Boolean Expression with two additional conditions

Problem: Given that $$ A \cdot B = 0$$ and $$A + B = 1$$ use algebraic manipulation to prove that: $$ (A + C) \cdot( \overline A + B ) \cdot (B + C) = B \cdot C$$ Answer: First, I will rewrite the ...
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Digital Logic Boolean Algebra

I am having a hard time understanding why A'+B' is not equal to the following expression: I understand that when I apply the 4 different combinations: 00,01,10,11 to A and B I get different results ...
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Sum-of-Product understanding

I am trying to learn Sum-of-Products and minterms and maxterms for logic design, but I don't know how the book gets the f(x1,x2,x3) part. I think I understand the minterm from the first pic (it's ...
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K-map: how to decide which implicant to be prime

I am a bit confused about deciding which implicants to turn into prime. If I group implicants 1 and 2 into a bigger implicant, can I group implicants 2 and 3 as well? For example, if I have the K-map ...
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What is meant by taking dual of a boolean expression?

I read online that if we have a set of SOP terms giving a boolean expression, then the POS terms of the complement of the SOP terms will give the same expression. POS(f)=SOP(f'). This is called ...
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How is the boolean expression for mux2 and mux4 deduced?

I've been looking at the following article: https://www.electronics-tutorials.ws/combination/comb_2.html I am confused as to how the boolean expression Q = a'b'A + ab'B + a'bC + abD is deduced for ...
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Realizing a function using only NAND-gates. Realize product of sums using NAND-gates

I have the function \$ f(x,y,z) = (xy'z' + x'yz' + xyz + x'y'z)' \$. I believe this can be written as: \$f(x,y,z)=(xy'z')' \cdot (x'yz')' \cdot (xyz)' \cdot (x'y'z)' \$ though I'm not sure. The ...
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Simplifying a logic expression with 3 variables

Below is a problem that I did. I am hoping that somebody can verify that my work is correct or tell me where I went wrong. Thanks. Problem: Simplify the following expression using Boolean Algebra: $...
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Simplifying a Boolean expression that has 3 variables in it

Problem: Simplify the following expression using Boolean Algebra: $$ z = (B + \overline C)(\overline B + C) + \overline{ \overline A + B + \overline C} $$ Answer: \begin{align*} z &= (B + \...
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2answers
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Trying to simplify a Boolean expression

Simplify the following expression using Boolean Algebra: $$ x = \bar{A} \bar{B} \bar{C} + \bar{A}BC + ABC + A \bar{B} \bar{C} + A \bar{B} C $$ Answer: \begin{align*} x &= \bar{A} \bar{B} \bar{C} +...
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Boolean Expression Minimization Question

How does one simplify this expression further? f(a,b,c,d) = a'b' + b'c' + b'd' + a'cd' + bcd The simplest form is f(a,b,c,d) = a'c + b'c' +b'd' + bcd I've been staring at it for a while and I don'...
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How to optimise a logic circuit? [closed]

I have the following 15 sets of binary outputs: ...
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Why does F + F' = 1?

I have the function: \$f(x,y,z,w) = wx + yz\$ I found its complement function to be: \$f '(x,y,z,w) = w'y' + w'z' + x'y' + x'z'\$ I have to show that: \$f + f '=1\$ but I can't see how to do it. ...
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How come (x*y)+(x*z')+(y*z) = (x*z')+(y*z)?

As the title says, I just can't figure this one out. I started with the initial function: f(x,y,z) = (x'yz) + (xy'z') + (xyz') + (xyz) and I have simplified it all the way down to this: f=(xy)+(xz')...
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Deriving PU / PD given a sketch of a PMOS

For the PMOS given below I can derive the function f, such that f inverted in its variables corresponds to the expression of PMOS(f) and f inverted equals NMOS(f). For this specific problem I have ...
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1answer
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Using NAND gates to construct OR/AND gates

I have this Boolean equation B'*C'*D' + A*C*D + C*D*E' and I was just wondering how to use nand gates to express this equation. With the schematic the inputs are NAND1 it is B'*C'*D' NAND ...
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use minimum number of NAND gates to realize this boolean expression [closed]

How should I proceed to find the minimum number of 2 input NAND gates to realize this boolean expression. I am allowed to use both complemented and non-complemented inputs. $$F = X.Y + Y.Z + \...
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Is there an intuitive reason for why NAND gate is a universal gate?

Now I know the maths and logic to figure out that every boolean function can be expressed using only AND and NOT gates, which in turn can be expressed using only NAND gate and hence every boolean ...
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1answer
60 views

Absolute difference with logical operators only

I'm trying to implement in a FPGA the test formula: abs(a-b)>1 a and b are unsigned 3 bits (0 to 7). The truth table is as follow: The test is: ...
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3answers
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What is the significance of the OR gate in this Full Adder circuit?

simulate this circuit – Schematic created using CircuitLab I don't understand the significance of the OR gate in the Full Adder because I don't think its input AND3 and AND2 will both ever be 1....
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I am stuck at this step C(AB+A'B')(simplify using boolean laws)

I cant figure out is AB+A'B' =1 or another answer
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Writing Truth Table Given Function

so I am taking a summer course and the teacher is not so helpful at the moment so I was thrown the following question to do: Derive a truth table for the following function: \$f(x,y,z)=\sum m(1,3,5,7)...
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Truth Table does not exhaust all combination. Can it be translated to a circuit? Is it ill-pose?

Step|A|B|C ---------- 1 |0|0|1 2 |1|0|1 3 |1|1|0 4 |1|0|0 From the truth table above, I can not find the equivalent boolean expression hence I can not ...
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Are these tables the same?

Is it possible from the table in the figure to go back to the circuit? (without know the circuit) sorry for my bad english The table of the first and second images are the same, however, written ...
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Asynchronous Sequencial circuit homework problem

i have some problem with this homework, i cant figure out how to continue the project. I would be grateful if you could help me (Sorry for my bad english) "Given the table in the figure, continue the ...
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1answer
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Design a synchronous sequential circuit

I can't figure out how to do this exercise: (sorry for my bad english) Design a synchronous sequential network according to the Mealy model with an input and an output which must assume the value 1 ...
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How can i create a boolean expression using a 4 x 1 mux?

I have got a question that asks Consider the Boolean expression \$F(A,B,C) = \bar{A} \bar{B} \bar{C}+ AB+ AC\$ . Implement this logic expression using one 4x1 multiplexer. Do not use \$C\$ as an input ...
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1answer
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3 Input XNOR gate made with XOR and not gates [duplicate]

hi i have implemented this equation with XOR and Not gate but i'm not getting the correct answers (A.B.C)+(A'.B'.C') What is the Problem ? thank you
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1answer
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Finding functions for JK / D / T flip flops

I am trying to understand this concept of flip flops. Given some Karnaugh Map, all I need to know is how to find the functions of various flip-flop types: sr flip-flop (\$s = \text{ ??} \quad r = \...
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1answer
45 views

expressing a boolean function in another way

The following boolean expression is given: g(p,q,r) = ((p->q') -> r) Depict this boolean function only by using the operations ∧ (AND), v (OR), ¬ (NEGATION). How can I do this? I thought I could ...
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Disjunctive Normal Form

The boolean expression: y= (a'+b) * (b(c'+d')) = (a'+b) * (bc' + bd') is given. Find out the disjunctive normal form to this boolean expression. My suggestion: y= a'bc'd' + a'bc'd + a'bcd' + abc'...
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The boolean expression from a circuit

Would the final boolean expression to the circuit be right? Or do I have to multiply the brackets b(c*d)' out to bc' + bd' simulate this circuit – Schematic created using CircuitLab Edit: My ...
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1answer
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logic gates, implementation [duplicate]

Task: Draw an AND circuit with 8 inputs, a circuit which implements the expression a∧b∧c∧d∧e∧f∧g∧h. Condition: Use only NOR-Gates with two inputs to solve this task. Could anyone give me a hint, ...
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1answer
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Boolean algebra implementation question [closed]

We could use boolean algebra to analyse the digital circuit and use boolean simplification to optimise the circuit. In boolean expression "1 + A" is 1, whatever A is 1 or 0. But in the real digital ...
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1answer
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Exclusive OR Proof

I’m given a question and its to prove exclusive OR can be the equivalent of using a NAND, OR and a AND gate show in the Boolean equation. $$A \oplus B = (A+B) \cdot (\overline{AB})$$ But when I look ...
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1answer
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Mosfet switch implementation

This is the second part of the problem which I try to solve from textbook. I wrote the truth table. Z is 1 only if A, B and C are 0. I've found a boolean equation like that : \$ \large Z = \bar {A} ...
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boolean algebra question regarding how to simplify a 5 input circuit

Background: I'm teaching myself computer architecture because I do not have a technical degree and want to learn this even though my finances won't ever allow me to have a formal education. I am doing ...
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1answer
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karnaugh maps simplification

I'm in the process of making a JK arbitrary sequence counter and I'm now making the k maps to get a logic equation. I was just wondering if this is allowed (looping the bits around the middle bit) as ...
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79 views

Implementation of boolean function using multiplexer

Is the following question correct? Consider the Boolean expression F(A,B,C) = ABC+ AB+ AC. Implement this logic expression using one 4x1 multiplexer. Do not use C as an input to the selectors of the ...
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2answers
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Simplification of the boolean expression [closed]

The following is the logic circuit: I have to simplify the following: (((AB)')'+(B+C)+(AB)'(B+C)')C =(AB+B+C+(A'+B')(B'C'))C =(B+C+A'B'C'+B'C')C =BC+C+A'B'C+B'C =C+A'BC'+B'C
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Implementing logic expression and the truth table of logic function

(a) Expression for \$Z\$ \$Z=(B+\overline{C})A+B(C+\overline{D})+BD\$ \$Z=AB+A\overline{C}+BC+B\overline{D}+BD\$ \$Z=AB+\overline{A}C+B\$ \$Z=B+A\overline{C}\$ (b) Truth table ...
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1answer
31 views

Checking overlow after addition of 2 integers in 2's complement form

Q) Let an−1an−2...a0 and bn−1bn−2...b0 denote the 2’s complement representation of two integers A and B respectively. Addition of A and B yields a sum S=sn−1sn−2...s0.The outgoing carry generated at ...
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Is this a specific logic function?

I mean it this fits a logic function or its the way its reduced only. This is the truth table \begin{array}{|c|c|c|c|} \hline A & B & C & V\\ \hline \hline \hline 0 & 0 & 0 & ...
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4 bit to 4 bit subtractor logic gates problem

I'm trying to create a 4 bit by 4 bit subtractor here but I get wrong output. In the picture it shows that 0000-1111=10001 which should be 0000-1111=11111. Please help me. By the way the rightmost IC ...