Maxterm: misinformation or different approach to understanding

Question

I have found myself in a crossway of two different explanations regarding Boolean algebra, Maxterm. However, I cannot grasp, whether there is a trick to it or whether one of explanations is an actual misinformation.

Here is a truth table for Minterms and Maxterms (table was found by searching it though the web):

This is another truth table for Maxterms only, however the order of indexes (\$M_0,..., M_7 \$) is exactly opposite from the table above (this table was provided by my university professor):

As I asked him for explanation, he answered that in the end, order of indexes doesn't matter, as long as it is true, that DeMorgan theorem applies to Minterms and Maxterms.

However I don't understand the relation between Minterms and Maxterms regarding DeMorgan's theorem.

The problem occurs when one asks for sum-of-products regarding a logic function given in a Maxterm form. For example, a function $$ f(x_1,x_2,x_3)=\prod M(1, 3, 6) $$ could be written as (regarding upper truth table)

$$ f(x_1,x_2,x_3) = (x_1+x_2+\bar x_3)\cdot(x_1+\bar x_2+\bar x_3)\cdot(\bar x_1+\bar x_2+x_3) $$

but could also be written as (regarding lower truth table)

$$ f(x_1,x_2,x_3) = (\bar x_1+\bar x_2+ x_3)\cdot(\bar x_1+x_2+x_3)\cdot(x_1+x_2+\bar x_3) $$

If I am not mistaken, only one of logic functions given in sum-of-product form, is correct. Here lies the confusion I cannot explain myself and would like to hear, what others think about this matter. If possible, one could explain it though Boolean algebraic expression.

Answer 1

Indeed, the ordering of indices doesn't matter in that $$ΠM(1,3,6) = Σm(1,3,6)$$ holds both for the straight and for the reversed orders of indexing, providing these are identical on the left and on the right of the equality sign.

The value of $$f(x_1, x_2, x_3)$$ does depend on the index ordering.