Product of Sums

Product of sums, step by step

Take this function:

Product of sums, step by step:

• Find every row where $Y = 0$. Here that is just $AB = 00$.
• For that row, write a sum (an OR) that equals 0 only there: $00$ gives $(A + B)$.
• AND all the sum terms together. With one term, that is simply $Y = A + B$.
• Check it: $A + B$ is 0 only when both are 0, and 1 for the other three rows. Correct.

OR gates into an AND

This circuit is the product of sums $Y = (A + B)\cdot(\overline{A} + C)$: two sum (OR) terms feeding one product (AND).

Try it:

1. Set a = 0, b = 0. The first sum goes to 0, and the AND forces y low no matter what c does.
2. Set a = 1, c = 0. Now the second sum is 0, and y is low again.
3. Notice the shape: OR gates on the left, one AND gate collecting them on the right. This is the mirror of the sum-of-products circuit.

Whichever is smaller

SOP and POS always produce the same function, so pick the one with fewer terms. That depends on the truth table:

• A table with mostly 0s has few 1-rows, so sum of products has fewer product terms. SOP wins.
• A table with mostly 1s has few 0-rows, so product of sums has fewer sum terms. POS wins.

The function in step 3 had three 1s and one 0. POS captured it in a single term, $A + B$, while SOP would have needed three products before simplification. When the output is mostly high, build from the zeros.