# Sum of products expression using Karnaugh maps

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

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!)

• Thanks a lot!This was very helpful! I was able to get the expression ¬A¬C+¬AB+AB using the K-map – Jarvis Ferns Apr 3 at 9:08
• @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 – Bram28 Apr 3 at 12:20
• Thanks for reminding that as I was able to conclude it! – Jarvis Ferns Apr 3 at 12:37