Why are products called minterms and sums called maxterms?

Question

Do they have a reason for doing so? I mean, in the sum of minterms, you look for the terms with the output 1; I don't get why they call it "minterms." Why not maxterms because 1 is well bigger than 0?

Is there a reason behind this that I don't know? Or should I just accept it without asking why?

Answer 1

The convention for calling these terms "minterms" and "maxterms" does not correspond to 1 being greater than 0. I think the best way to answer is with an example:

Say that you have a circuit and it is described by X̄YZ̄ + XȲZ.

"This form is composed of two groups of three. Each group of three is a 'minterm'. What the expression minterm is intended to imply it that each of the groups of three in the expression takes on a value of 1 only for one of the eight possible combinations of X, Y and Z and their inverses." http://www.facstaff.bucknell.edu/mastascu/elessonshtml/Logic/Logic2.html

So what the "min" refers to is the fact that these terms are the "minimal" terms you need in order to build a certain function. If you would like more information, the example above is explained in more context in the link provided.

Edit: The "reason they used MIN for ANDs, and MAX for ORs" is that:

In Sum of Products (what you call ANDs) only one of the minterms must be true for the expression to be true. In Product of Sums (what you call ORs) all the maxterms must be true for the expression to be true.

Answer 2

min(0,0) = 0
min(0,1) = 0
min(1,0) = 0
min(1,1) = 1

So minimum is pretty much like logical AND.

max(0,0) = 0
max(0,1) = 1
max(1,0) = 1
max(1,1) = 1

So maximum is pretty much like logical OR.

Answer 3

I believe that AB is called a minterm is because it occupies the minimum area on a Venn diagram; while A+B is called a MAXTERM because it occupies a maximum area in a Venn diagram. Draw the two diagrams and the meanings will become obvious Ed Brumgnach

Answer 4

In Sum Of Products (SOP), each term of the SOP expression is called a "minterm" because,

say, an SOP expression is given as: F(X,Y,Z) = X'.Y'.Z + X.Y'.Z' + X.Y'.Z + X.Y.Z

for this SOP expression to be "1" or true (being a positive logic), ANY of the term of the expression should be 1. thus the word "minterm".

i.e, any of the term (X'Y'Z) , (XY'Z') , (XY'Z) or (XYZ) being 1, results in F(X,Y,Z) to be 1!! Thus they are called "minterms".

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On the other hand, In Product Of Sum (POS), each term of the POS expression is called a "maxterm" because,

say an POS expression is given as: F(X,Y,Z) = (X+Y+Z).(X+Y'+Z).(X+Y'+Z').(X'+Y'+Z)

for this POS expression to be "0" (because POS is considered as a negative logic and we consider 0 terms), ALL of the terms of the expression should be 0. thus the word "max term"!!

i.e for F(X,Y,Z) to be 0, each of the terms (X+Y+Z), (X+Y'+Z), (X+Y'+Z') and (X'+Y'+Z) should be equal to "0", otherwise F won't be zero!!

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Thus each of the terms in POS expression is called a MAXTERM (maximum all the terms!) because all terms should be zero for F to be zero, whereas any of the terms in POS being one results in F to be one. Thus it is known as MINTERM (minimum one term!)

Answer 5

Here is another way to think about it.

A product is called a minterm because it has minimum-satisfiability where as a sum is called a maxterm because it has maximum-satisfiability among all practically interesting boolean functions.

They are called terms because they are used as the building-blocks of various canonical representations of arbitrary boolean functions.

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Details:

Note that '0' and '1' are the trivial boolean functions. Assume a set of boolean variables x1,x2,...,xk and a non-trivial boolean function f(x1,x2,...,xk).

Conventionally, an input is said to satisfy the boolean function f, whenever f holds a value of 1 for that input.

Note that there are exactly 2^k inputs possible, and any non-trivial boolean-function can satisfy a minimum of 1 input to a maximum of 2^k -1 inputs.

Now consider the two simple boolean functions of interest: sum of all variables S, and product of all variables P (variables may/may-not appear as complements). S is one boolean function that has maximum-satisfiability hence called as maxterm, where as P is the one having minimum-satisfiability hence called a minterm.