How Can You Solve Boolean Expressions Using Only NAND and NOR Gates?

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In summary: Q2: Design a combinational logic circuit that converts a 4 bit sign magnitude representation of a number to a 4 bit 2-s complement representation.Q3: Suppose you require a 2 bit adder circuit. That is, a binary number xy is to be added to a binary number uv in order to yield a binary number abc. Design such an adder circuit using three 16:1 multiplexors. Show how the circuit can also be designed using three 8:1 multiplexors.In summary, the first question asks to represent given expressions using only NAND and NOR gates, while the second question involves designing a combinational logic circuit that converts a 4 bit sign magnitude representation of a number to a
  • #1
Berbanog
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Edited Q1 + solution attempts

Homework Statement



Q1: Represent the following using only NAND gates, and only NOR gates
Q1a) A.B + ~(A.C).~(B+C)
Q1b) (A XOR B) + ~(A XOR B).(B + C)

Q2: Design a combinational logic circuit that converts a 4 bit sign magnitude representation of a number to a 4 bit 2-s complement representation.

Q3: Suppose you require a 2 bit adder circuit. That is, a binary number xy is to be added to a binary number uv in order to yield a binary number abc. Design such an adder circuit using three 16:1 multiplexors. Show how the circuit can also be designed using three 8:1 multiplexors.

Homework Equations


Far as I'm concerned there aren't much 'equations' to write...

The Attempt at a Solution


Q1a) This was answer i got = ~(~A+~B~(~A+~C)+~(~B.~C))
To solve it, i worked backward and started with NAND/NOR gate over the entire equation and found the variable that fits in it that is equal to the original equation.
Q1b) ~(A XOR B) = ~A XOR B <-- i believe i needed to use this to solve this question, but don't think i was getting any closer to getting the solution to this...
Q2 + Q3: I have no clue what these questions are trying to ask...! If someone could give any tips/sites that may help, i will appreciate it very much :D

no need to hurry since I'm still solving them myself, but questions difficult to solve always generate headaches!
 
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  • #2
Berbanog said:

Homework Statement



i put brackets to represent the bar on top of variables
(A.C) = (A.C)
[A.C] = A'+C'
and XOR sign been replaced with '(X)'

Q1: Represent the following using only NAND gates, and only NOR gates
Q1a) A.B + [A.C].[B+C]
Q1b) A (X) B + [A (X) B](B + C)

Q2: Design a combinational logic circuit that converts a 4 bit sign magnitude representation of a number to a 4 bit 2-s complement representation.

Q3: Suppose you require a 2 bit adder circuit. That is, a binary number xy is to be added to a binary number uv in order to yield a binary number abc. Design such an adder circuit using three 16:1 multiplexors. Show how the circuit can also be designed using three 8:1 multiplexors.

Homework Equations


Far as I'm concerned there aren't much 'equations' to write...

The Attempt at a Solution


Q1a) This was answer i got = [A'+B'[A'+C']+[B'C']]
To solve it, i worked backward and started with NAND/NOR gate over the entire equation and found the variable that fits in it that is equal to the original equation.
Q1b) [A (X) B] = A' (X) B <-- i believe i needed to use this to solve this question, but don't think i was getting any closer to getting the solution to this...
Q2 + Q3: I have no clue what these questions are trying to ask...! If someone could give any tips/sites that may help, i will appreciate it very much :D

no need to hurry since I'm still solving them myself, but questions difficult to solve always generate headaches!

Welcome to PhysicsForums!

Regarding your first question, I have a hard time deciphering what's going on. Here's a tip: you can use inline [iline]\LaTeX[/iline] with the [iline ] tag (switch to advanced composition and click on the little sigma button to bring up the LaTeX reference.) However, standard (VHDL) notation is to represent a NOT using a forward slash (\) or a tilde (~) in front of something, like \A or ~(A&B). An OR is just the plus sign, and an AND is, of course, just an &.

So, maybe I'm misreading but, I believe at one point, you use ~(A & B) (A NAND B) and simplify down to ~A & ~B (NOT A AND'ed with NOT B), but these do not commute this way. If you continue reading in your textbook, you'll find that NAND and NOR can be used to build every other type of gate. You can go in the forwards direction by replacing the unitary / binary operators with NANDs and NORs instead, and trying to simplify:
http://en.wikibooks.org/wiki/Digital_Circuits/NAND_Logic
http://en.wikibooks.org/wiki/Digital_Circuits/NOR_Logic

For Question 2, you'll need to do some additional reading in your textbook on sign formats, and sign conversions. Basically, you want to design a so-called black box that takes an input, in sign + magnitude representation, and then outputs the number in 2's complement. You can start with the Wikipedia article, but you should read through the pertinent textbook sections:
http://en.wikipedia.org/wiki/Signed_number_representations

For Question 3, they want you to use multiplexors (and simple logic gates) to realize mathematical functions. You want your inputs going to the given multiplexers in such a way that the output happens to be the sum of the inputs. To get you started on that topic:
http://www.play-hookey.com/digital/adder.html
http://www.play-hookey.com/digital/multiplexer_two_input.html
http://www.play-hookey.com/digital/decoder_demux_two.html
 
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  • #3
*Edit! As 'MATLABdude' wanted i have edited my questions into the correct format

Q1: Represent the following using only NAND gates, and only NOR gates
Q1a) A.B + ~(A.C).~(B+C)
Q1b) (A XOR B) + ~(A XOR B).(B + C)
 

FAQ: How Can You Solve Boolean Expressions Using Only NAND and NOR Gates?

1. What is Boolean logic and how is it used in circuit design?

Boolean logic is a mathematical system used to represent and manipulate true and false values, often denoted as 1 and 0. In circuit design, Boolean logic is used to create logical circuits that perform operations based on these binary values. This allows for the creation of complex logic functions in a simple and efficient manner.

2. What are logic gates and how do they work?

Logic gates are fundamental building blocks of digital circuits that perform logical operations on one or more binary inputs to produce a single binary output. They are represented by symbols and can be combined in different ways to create complex logic functions. The output of a logic gate depends on the values of its inputs and the specific logic function it performs.

3. What are the basic types of logic gates?

The three basic types of logic gates are AND, OR, and NOT gates. The AND gate produces a 1 output only when both inputs are 1. The OR gate produces a 1 output if either or both inputs are 1. The NOT gate produces a 1 output when the input is 0, and vice versa. These gates can be combined to create more complex logic functions.

4. How is Boolean logic used to design sequential circuits?

Sequential circuits are digital circuits that utilize memory elements to store information and produce outputs based on both current and previous inputs. Boolean logic is used to design the logic functions within these circuits, which determine the behavior of the memory elements and the overall operation of the circuit.

5. What is the difference between combinational and sequential logic?

Combinational logic circuits produce outputs based solely on the current inputs, while sequential logic circuits use both current and previous inputs to produce outputs. Combinational logic circuits do not have memory elements, while sequential logic circuits do. This means that sequential circuits can store and process information, while combinational circuits can only process information in real-time.

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