I came across following fact:
If we AND two functions expressed as CDNF (Canonical Disjunctive Normal Form), then the result contains sum of commont minterms.
For example consider two functions $f_1$ and $f_2$ on three variables $A,B$ and $C$ expressed in CDNF. Then they will have eight minterms: $m_0$ to $m_7$.
If
$f_1 = \sum ({m_1,m_2,m_3}) = m_1+m_2+m_3$ and
$f_1 = \sum ({m_2,m_3,m_4})=m_2+m_3+m_4$
then,
$f_1.f_2=m_2+m_3$ as $m_2$ and $m_3$ are common in both $f_1$ and $f_2$.
However now I am thinking about what will be the result of
$f_1+f_2$
If we consider $F_1$ and $F_2$ on three variables $A,B$ and $C$ expressed in CCNF (Canonical Conjuctive Normal Form) as follows:
$F_1 = \prod ({M_1,M_2,M_3}) = M_1.M_2.M_3$ and
$F_1 = \prod ({M_2,M_3,M_4}) = M_2.M_3.M_4$
then what will be the result of
$F_1.F_2$
$F_1+F_2$
After some thinking, I came up with following:
$f_1+f_2$ will be
union of minterms in both functions, that is sum of all minterms with the common minterm appearing only once.
Thus, $f_1+f_2 = m_1+m_2+m_3+m_4$
$F_1.F_2$ will also be
union of the maxterms in both functions. That is the product of all maxterms with the common maxterms appearing only once.
Thus, $F_1.F_2=M_1.M_2.M_3.M_4$
I found it very tricky to think about $F_1+F_2$. In the end I felt it should be
AND of OR of two different maxterms with no OR of same and common maxterms.
$\therefore F_1+F_2= (M_1+M_2)(M_1+M_3)(M_1+M_4)(M_2+M_3)(M_2+M_4)(M_3+M_4)$ So $(M_2+M_2)$ and $(M_3+M_3)$ will not be there.
as $f_1.f_2$ was
OR of AND of two same and common minterms with no AND of different minterms.
So above $f_1.f_2$ was actually $m_2.m_2+m_3.m_3$. This because AND of different terms will be $0 (false)$ and $0+a=a$
However, if I am correct with $(M_2+M_2)$ and $(M_3+M_3)$ being not part of $F_1+F_2$, I don't understand why that would be. If I think as dual of "AND of different terms will be $0 (false)$ and $0+a=a$", it would be "OR of same terms will be $1 (true)$ and $1.a=a$". But I dont get how OR of same terms will be 1?
I am just trying to find some pattern here and didnt found this explained anywhere.
Q. Am I correct with these three guessings?
Q. If yes then I will like to know about above bold-faced question: how OR of same terms will be 1?