# How to convert between Sum Of Products and Product of sums?

I have a Boolean expression. we'll call it F.

for instance, F = ab' + ad + c'd + d'.

Assuming I did all the necessary steps too get F complement , i.e. F'.

I got: F' = b'd + ac'd'.

How do I get the Product of sums form of F?

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$$F=(F')'=(b'd+ac'd\,')'=(b'd)'(ac'd\,')'=(b+d\,')(a'+c+d)\;.$$

(Note: I did not check your $F'$.)

Because of the way the De Morgan laws work, the complement of a product of sums is always a sum of products, and the complement of a sum of products is always a product of sums.

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Thank you, but that is exactly what I've got (on my paper...) and it doesn't work .. are you sure it's right? –  Billie Feb 3 '13 at 8:25
@user1798362: I’m sure that it would be right if your $F'$ were right. The problem is that your $F'$ is wrong: I get $a'b'cd+ab'c'd'$ for $F'$. –  Brian M. Scott Feb 3 '13 at 20:30