How does the product of sum form work? I understand that you take the product of the sums of the negated inputs of all rows of the truth table whose output is false. Taking XOR for example:
| A | B | Output |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
The product of sum form would be:
$(\bar A + \bar B)(A + B)$
However, I don't understand why this method yields the correct boolean equation?
I understand why the sum of product form works but the product of sum form seems quite unintuitive to me.
Considering your example, we have the logical expression:
Using similar logic to the sum of product form, we know that:
${(\bar A \bar B) + (AB)} = F$
Therefore:
$\overline{(\bar A \bar B) + (AB)} = T$
With continuous application of Demorgan's Theorem ($\overline{X+Y} = \bar X \bar Y$):
$\overline{(\bar A \bar B)} \overline{(AB)} = T$
$(\bar{\bar A } + \bar{\bar B}) (\bar A + \bar B) = T$
$(A + B) (\bar A + \bar B) = T$
As desired.