Minterms and Maxterms in Boolean Algebra

Any Boolean function or logical expression can be expressed in either canonical/standard sum of products form or canonical/standard product of sums form. The standard sum of products form of a logical expression contains different product terms which are added together, and each product term is referred to as a minterm. On the other hand, the standard product of sums form of a logical expression contains different sum terms which are multiplied together, and each sum term is called a maxterm. In this article, we will discuss about the minterm and max terms.

What is Minterm?

When a Boolean function or logical expression is expressed in the SSOP (Standard Sum of Product) Form or canonical form, then each term of the expression is called a minterm.

In other words, a product term of a logical expression in n variables, which contains each of the n variables in its either complemented or un-complemented form is called a minterm.

A minterm is often represented as mi, where, i is an integer in between 0 and 2^(n-1). Here, "n" is the number of variables in the expression. Therefore, minterms can be denoted as m₀, m₁, m₂,m₃,… Here, the suffixes the decimal codes of the combinations of variables.

In a minterm, a variable will appear in its complemented form if its value is equal to 0. And, the variable will appear in its un-complimented form if its value is equal to 1.

Now, let us consider some example to understand how a logical expression is expressed in minterms.

For a logical expression in 2-variables (A and B), the possible minterms are,

$$\mathrm{m_0=\overline{A}\overline{B}}$$

$$\mathrm{m_1=\overline{A}B}$$

$$\mathrm{m_2=A\overline{B}}$$

$$\mathrm{m_3=AB}$$

For a logical expression in 3-variables (A, B, and C), the possible minterms are,

$$\mathrm{m_0=\overline{A}\:\overline{B}\:\overline{C}}$$

$$\mathrm{m_1=\overline{A}\:\overline{B}C}$$

$$\mathrm{m_2=\overline{A}B\overline{C}}$$

$$\mathrm{m_3=\overline{A}BC}$$

$$\mathrm{m_4=A\overline{B}\:\overline{C}}$$

$$\mathrm{m_5=A\overline{B}C}$$

$$\mathrm{m_6=AB\overline{C}}$$

$$\mathrm{m_7=ABC}$$

Here, we can see that a logical function in two variables has four (2² = 4) minterms, and the logical function in 3-variables has eight (2³ = 8) minterms. The variable in complemented form (represented with a bar over the variable) has a value equal to 0 and the variable in un-complemented form has a value equal to 1.

What is Maxterm?

When a Boolean function or logical expression is expressed in the SPOS (Standard Product of Sum) Form or canonical form, then each term of the expression is called a maxterm.

In other words, a sum term of a logical expression in n variables, which contains each of the "n" variables in its either complemented or un-complemented form is called a maxterm.

The maxterm is often represented by Mi, where "i" is an integer between 0 and 2^(n-1). Here, "n" is the total number of variable in the logical expression. Therefore, maxterms of a logical expression can be denoted as M₀, M₁, M₂,… where the suffixes represent their decimal codes of the combinations.

In the case of maxterms, a variable will be written in its complemented form if its value is equal to 1, and the variable will be written in its un-complemented form if its value is equal to 0.

Now, let us know how we can express a logical function in the form of maxterms.

For a Boolean function in 2 variables (A and B), the possible maxterms are,

$$\mathrm{m_0=\lgroup A+B\rgroup}$$

$$\mathrm{m_1=\lgroup A+\overline{B}\rgroup}$$

$$\mathrm{m_2=\lgroup\overline{A}+B\rgroup}$$

$$\mathrm{m_3=\lgroup\overline{A}+\overline{B}\rgroup}$$

For a Boolean expression in 3 variables (A, B, C), the possible maxterms are,

$$\mathrm{m_0=\lgroup A+B+C\rgroup}$$

$$\mathrm{m_1=\lgroup A+B+\overline{C}\rgroup}$$

$$\mathrm{m_2=\lgroup A+\overline{B}+C\rgroup}$$

$$\mathrm{m_3=\lgroup A+\overline{B}+\overline{C}\rgroup}$$

$$\mathrm{m_4=\lgroup\overline{A}+B+C\rgroup}$$

$$\mathrm{m_5=\lgroup\overline{A}+B+\overline{C}\rgroup}$$

$$\mathrm{m_6=\lgroup\overline{A}+\overline{B}+C\rgroup}$$

$$\mathrm{m_7=\lgroup\overline{A}+\overline{B}+\overline{C}\rgroup}$$

Here, from these two logical expressions in 2-variables and 3-variables respectively, we can see that a logical function in two variables has four (2² = 4) maxterms, and the logical function in 3-variables has eight (2³ = 8) maxterms. In this case, the variable in un-complemented form (represented with a bar over the variable) has a value equal to 0 and the variable in complemented form has a value equal to 1.

Conclusion

This is all about minterms and maxterms in Boolean algebra. From the above discussion, we may conclude that a minterm is a product term of a logical expression, when the expression is represented in its standard sum of product (SSOP) form. On the other hand, a maxterm is a sum term of a logical expression, where the logical expression is expressed in the standard product of sums (SPOS) form.

The common point about both minterm and maxterm is that they contain each of the "n" variables of the logical function.