I understand that in terms of minterms,

F (Boolean Function) = Sum of Products and thus will yield true when either of the products is true.

But I am unable to develop any intuition for maxterms,

F(Boolean Function) = Product of Sums (of maxterms).

Is F = Product of sums just a different way of representing F = Sum of Products?

If yes, is it possible to prove that using the duality principle?


Indeed, maxterm is the concept dual to minterm. As you mention, $$ f(x) = \bigvee_{y\colon f(y)=1} \text{"$x=y$"} = \bigvee_{y\colon f(y)=1} \bigwedge_{i=1}^n \text{"$x_i=y_i$"}. $$ Here $\vee$ is OR (your addition), $\wedge$ is ADD (your multiplication), the input is $x = x_1,\ldots,x_n$, and "$x_i=y_i$" is either $x_i$ (if $y_i$ is True) or $\bar{x}_i$ (if $y_i$ is False).

Similarly, we have $$ f(x) = \bigwedge_{y\colon f(y)=0} \text{"$x \neq y$"} = \bigwedge_{y\colon f(y)=0} \bigvee_{i=1}^n \text{"$x_i \neq y_i$"}. $$ This time "$x_i\neq y_i$" is $x_i$ if $y_i$ is False, and $\bar{x}_i$ if $y_i$ is True.

When the function is monotone, we can improve on the first representation: $$ f(x) = \bigvee_{\text{$y$ minterm}} \text{"$x \geq y$"} = \bigvee_{\text{$y$ minterm}} \bigwedge_{i\colon y_i=1} x_i. $$ Here "$x \geq y$" means $x_i \geq y_i$ for all $i$, and a minterm is a satisfying assignment of $f$ which is minimal in the sense that changing any 1 to 0 changes it to an unsatisfying assignment.

We can define maxterms analogously – maximal unsatisfying assignments, and then $$ f(x) = \bigwedge_{\text{$y$ maxterm}} \text{"not $x \leq y$"} = \bigwedge_{\text{$y$ maxterm}} \bigvee_{i\colon y_i=0} x_i. $$

  • $\begingroup$ Is maxterm the complement of a minterm or not? $\endgroup$
    – Archer
    Sep 27 '18 at 8:02
  • 1
    $\begingroup$ No, it’s the dual concept to minterm. $\endgroup$ Sep 27 '18 at 13:42

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