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Use a Karnaugh Map to simplify the following Sum of Products expression, where A is the most significant bit of each component and C the least significant: Y = ∑(0,2,3,6,7).

The answer is suppose to be : Y = B+¬A¬C .But,I don't understand how to get solution.Can someone give me a hint or push me in the right direction? Thanks in advance

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OK, so first of all, I assume the numbers are represented in binary, i.e. 0=000, 7=111, etc. And $A$ being the most significant digit, it would be the first, and $C$ being the last, and thus $B$ being the middle digit. Thus, for example, $010$ would correspond to $A'BC'$

As such, $\Sigma(0,2,3,6,7)$ would be $000+010+011+110+111$, i.e. $A'B'C'+A'BC'+A'BC+ABC'+ABC$

Ok, and so now you need to simplify that ...

Using algebra, it's easy:

$A'BC'+A'BC$ simplifies to $A'B$ by Adjacency

Likewise, $ABC'+ABC=AB$

And then, $A'B+AB=B$

So, you get:

$A'B'C'+B$

Now there is a handy principle you can use:

Reduction

$A +A'B=A+B$ (the $A'B$ term 'reduces' to just $B$ given the term $A$)

And so using Reduction, your expressin becomes $A'C'+B$, which is indeed the desired answer

You, however, have to use a K-map... can you do that? (Note: If you do use a k-map, you'll quickly figure out how the Adjacency principle got its name!)

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  • $\begingroup$ Thanks a lot!This was very helpful! I was able to get the expression ¬A¬C+¬AB+AB using the K-map $\endgroup$ – Jarvis Ferns Apr 3 at 9:08
  • $\begingroup$ @JarvisFerns Then you're almost there! just note that you can combine the two pairs of cells corrsponding to A'B and AB into one cluster of four cells, corresponding to just B $\endgroup$ – Bram28 Apr 3 at 12:20
  • $\begingroup$ Thanks for reminding that as I was able to conclude it! $\endgroup$ – Jarvis Ferns Apr 3 at 12:37

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