How Can I Simplify This Boolean Expression Using XOR and XNOR Functions?

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Discussion Overview

The discussion revolves around simplifying a Boolean expression using XOR and XNOR functions, with the aim of ultimately drawing a logic circuit using NAND and XOR gates. Participants explore various methods for simplification, including truth tables and Karnaugh maps.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • One participant presents a Boolean expression and seeks help in simplifying it, expressing uncertainty about their progress.
  • Another suggests drawing a truth table to identify patterns that may correspond to XOR or NAND functions.
  • A participant proposes a modified expression and asks for validation on whether they are on the right track before further simplification.
  • There is a mention of a potential simplification using Karnaugh maps, although the method is not universally agreed upon as the easiest approach.
  • Multiple participants express similar expressions, indicating they are working through the same simplification process.

Areas of Agreement / Disagreement

Participants generally share similar approaches to the problem, but there is no consensus on the best method for simplification or the correctness of the proposed expressions. The discussion remains exploratory and unresolved.

Contextual Notes

Some expressions and simplifications presented depend on the assumptions made about the variables and their relationships. The discussion includes various attempts at simplification without a definitive resolution.

TheTopGun
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Homework Statement



Hey there, I'm having trouble simplifying a boolean expression using XOR and XNOR functions.
The final goal is to draw a logic circuit for the expression using NAND and XOR gates only.

Homework Equations



Assuming
W' = Not WW' X' Y' Z' + W' X' Y Z + W' X Y' Z + W' X Y Z' + W X Y' Z' + W X' Y' Z

The Attempt at a Solution

W' X' (Y⊕Z)' + W' X (Y⊕Z) + ...

So far if I'm working correctly (?) I can simplify the first four expressions, It's just the last two that get me as they have no like terms ?

Once I understand this I should have no troubles drawing the logic circuit.

Thankyou for any responses
 
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you could try drawing a truth table for the terms and see if you spot a pattern that matches XOR or NAND

like X xor Y'
 
Hmm okay thanks,

I've come up with this

W' X' (Y⊕Z)' + W' X (Y⊕Z) + W Y' (X⊕Z)

Is this on the right track..before I simplify further.
 
W x y' z' + w x' y' z = w y'❲ ... ❳

wow, PF has such a nanny editor that it converted my upper case to lower!
 
TheTopGun said:
Hmm okay thanks,

I've come up with this

W' X' (Y⊕Z)' + W' X (Y⊕Z) + W Y' (X⊕Z)

Is this on the right track..before I simplify further.

That seems right. Once complete you could do a truth table on it and see if it agrees with original.
 
TheTopGun said:
Hmm okay thanks,

I've come up with this

W' X' (Y⊕Z)' + W' X (Y⊕Z) + W Y' (X⊕Z)

Is this on the right track..before I simplify further.
You can see how to simplify this further? If not, I suggest that you construct a 4 variable Truth Table. Also construct the Truth Table for your simplified expression, and with any luck there may be some correspondence.

I haven't tried it.
 
For future reference (im not sure why anyone would care lol, but anyway), the equation can be simplified down to:

W' (X⊕Y⊕Z)' + Y' (W⊕X⊕Z)' ...where (A⊕B⊕C)' represents a XNOR gate.

This was obtained using karnough maps but I am sure there are easier ways...
 

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