Sum of Product Expansion

Question

Find sum of product expansion of Boolean function F(x, y, z) =x ̄y+ ̄xyz.

Answer 1

First, we make a truth table.

X Y Z Output

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0 0 0 0

0 0 1 0

0 1 0 0

0 1 1 1

1 0 0 1

1 0 1 1

1 1 0 0

1 1 1 0

Now, we find that the only times when our Output is high (1), is when it is x'yz + xy'z' + xy'z.

Then, we minimize this by noticing that the last two terms have xy', and both z and z' produce a high output.

This means we can remove the z component entirely, and reduce to:

x'yz+xy'.

This is our minimal SOP form. In a more difficult case, it would be advantageous to construct the Karnaugh Map:

z' z

x'y' 0 0

x'y 0 1

xy 0 0

xy' 1 1

You notice that our largest circles possible end up giving us the same answer as above--you eliminate the z value in the second term, and once again have the minimal SOP.