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I have to simplify the following boolean expression by proceeding only in POS form and not combining variables:

XYZD + X'P +Y'P + Z'P + D'P

Notation is

  • . means AND
  • + means OR
  • ' means NOT

"." is omitted wherever unambiguous as AND has precedence over OR.

I am able to solve this by combining terms but it is not allowed :

= XYZD + P.(X'+Y'+Z'+D')

= XYZD + P.(XYZD)' (DEMORGAN LAW)

= (XYZD + P) . ( XYZD+ (XYZD)' ) (Distributive Law A+B.C = (A+B).(A+C) )

= (XYZD + P) . (1) *(Complementary Law)

= XYZD + P (IDENTITY)

Why do this :

  1. This is generally useful when someone has already simplified an expression to pos using boolean laws and we don't want to go backwards
  2. if we are given an intermediate form of a boolean expression to proceed with, combining variables is counter-intuitive and there should be an alternative way to solve considering single variables at a time.

For eg : I should not get stuck when i have done the following :

Minimize : x.y.z.d+(x.y.z.d)'.p

= x.y.z.d+((x.y.z)'+d').p DEMORGAN Law

= x.y.z.d+((x.y)'+z'+d').p DEMORGAN Law

= x.y.z.d+(x'+y'+z'+d').p DEMORGAN Law

= x.y.z.d+x'.p+y'.p+z'.p+d'.p DISTRIBUTIVE Law

= x.y.z.d+x'.p+y'.p+z'.p+d'.p

now how to proceed further ???

Nishant
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    Welcome to math.se. Can you please explain what multiplication and addition represent? This is not standard notation. – Couchy Sep 09 '20 at 09:06
  • Please use Mathjax to render the math it will be more readable. – user577215664 Sep 09 '20 at 09:06
  • If multiplication is conjunction, addition disjunction, and equality is equivalence, then your third step (by distribution, complement, and identity) is incorrect. – Couchy Sep 09 '20 at 09:09
  • Hi all. Have edited the question as per your questions. The notation is standard boolean algebra. Also @Couchy have broken down the steps to make it simpler. I think it is correct. – Nishant Sep 09 '20 at 15:15
  • XYZD + X'P +Y'P + Z'P + D'P $\to$ XYZD + (X'+Y'+Z')P $\to$ (XYZD + X'+Y'+Z')(XYZD+P)$\to$ (XYZ +X'+Y'+Z')(D+X'+Y'+Z')(XYZ+P)(D+P)$\to\ldots$ – Couchy Sep 09 '20 at 16:03

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