# Intuition for maxterms

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 $$f(x) = \bigwedge_{\text{y maxterm}} \text{"not x \leq y"} = \bigwedge_{\text{y maxterm}} \bigvee_{i\colon y_i=0} x_i.$$