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Find the principal disjunctive normal form (PDNF) of a Boolean expression $$((p\wedge q) \rightarrow r)\vee((p\wedge q)\rightarrow \neg r).$$ I tried by expanding it but I am stuck with the expression $(\neg p \vee \neg q \vee r) \vee (\neg p \vee \neg q \vee \neg r)$. I don't know how to convert them into min terms. Please help me.

Random Jack
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Balaji
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    Your last expression can be written as $\text{Something}\lor (r\lor \neg r)$. – Git Gud Jan 03 '16 at 19:20
  • First you must see that either the equation is sum of products or not.As it has only disjunctions, so it is not possible to find DNF for this equation. – Muhammad Asif Dec 20 '16 at 21:05

2 Answers2

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Expression mentioned is a Tautology. $$((p.q) \rightarrow r)+((p.q)\rightarrow r')=((p'+q'+r)+(p'+q'+r'))=p'+q'+1 = 1$$ So PDNF corresponding to it is $$(p'.q'.r')+(p'.q'.r)+(p'.q.r')+(p'.q.r)+(p.q'.r')+(p.q'.r)+(p.q.r')+(p.q.r)$$

Novice
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The required normal form is ( if I am not wrong ):-

(p^q^r)v(~p^q^r)v(~p^~q^r)v(p^q^~r)

Once you get the form : (¬p∨¬q∨r)∨(¬p∨¬q∨¬r)
Treat this as (¬p)∨(¬q)∨(r)∨(¬p)∨(¬q)∨(¬r)
And convert the individual literals into minterms.

To convert any variable P to a minterm ,just add : ^(Qv~Q)^(Rv~R) to P
That is :P^(Qv~Q)^(Rv~R)
And then use the distributive laws.

Amrutesh Mishra
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