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):

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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):

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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.


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.

  • \$\begingroup\$ Any idea which order should be considered "right" one? \$\endgroup\$ – Keno Oct 16 '20 at 12:33
  • \$\begingroup\$ Any will do providing those are consistent throughout your calculations. Also, you can specify your convention in the text when writing your report/paper. \$\endgroup\$ – V.V.T Oct 16 '20 at 12:34
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    \$\begingroup\$ So, you are saying if I am going to stick with one way of indexing, I should do so always? So, I guess if my professor does it in reverse order, then even I should do so. \$\endgroup\$ – Keno Oct 16 '20 at 12:37
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    \$\begingroup\$ Hope they are not especially particular, but 'cause it doesn't matter, do as they please. \$\endgroup\$ – V.V.T Oct 16 '20 at 12:38
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    \$\begingroup\$ It would certainly cause less confusion if you adapt to the convention your main audience (who grades your work) uses. Nothing wrong with noting (and referencing) different conventions and your reasoning for choosing one. \$\endgroup\$ – user_1818839 Oct 16 '20 at 13:40

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