Digital-electronics 简明教程

Minterms and Maxterms in Boolean Algebra

任何布尔函数或逻辑表达式都可以用或运算标准积之和形式或乘运算标准和之积形式表达。逻辑表达式的或运算标准积之和形式包含不同的乘积项，它们被相加在一起，每个乘积项被称为一个最小项。另一方面，逻辑表达式的乘运算标准和之积形式包含不同的和项，它们被相乘在一起，每个和项被称为一个最大项。在本文中，我们将讨论最小项和最大项。

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?

当一个布尔函数或逻辑表达式用或运算标准积之和形式（标准形式）表达时，那么表达式中的每项被称为一个最小项。

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.

换句话说，一个逻辑表达式在 n 个变量中的乘积项，只要包含 n 个变量中每一个变元的补集或未补集形式，就称为一个最小项。

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.

一个最小项通常表示为 mi，其中 i 是一个介于 0 和 2(n-1) 之间的整数。此处，"n" 是表达式中的变量个数。因此，最小项可以表示为 m0、m1、m2、m3、……此处，后缀是变量组合的十进制代码。

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.

在一个最小项中，如果一个变量的值等于 0，则该变量将以其补集形式出现。并且，如果一个变量的值等于 1，则该变量将以其未补集形式出现。

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.

现在，让我们考虑一些例子来理解如何在最小项中表达逻辑表达式。

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

对于一个 2 变量（A 和 B）的逻辑表达式，可能的最小项是，

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}

对于一个 3 变量（A、B 和 C）的逻辑表达式，可能的最小项是，

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}

在这里，我们可以看到，两个变量中的一个逻辑函数有四个 (22 = 4) 个极小项，而 3 个变量中的逻辑函数有八个 (23 = 8) 个极小项。补码形式中的变量（用变量上的条形表示）具有等于 0 的值，非补码形式中的变量具有等于 1 的值。

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?

当布尔函数或逻辑表达式以 SPOS（和的标准积）形式或规范形式表示时，则表达式中的每个项称为 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.

换句话说，逻辑表达式中包含每个“n”个变量（以其补码或非补码形式包含）的和项称为 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.

极大项通常用 Mi 表示，其中“i”是 0 到 2(n-1) 之间的整数。此处，“n”是逻辑表达式中的变量总数。因此，逻辑表达式的极大项可以表示为 M0、M1、M2、……，其中后缀表示它们组合的小数代码。

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.

对于极大项，如果变量的值等于 1，则该变量将以其补码形式书写，如果变量的值等于 0，则该变量将以其非补码形式书写。

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.

现在，让我们了解如何以最大项的形式表达逻辑功能。

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

对于 2 个变量 (A 和 B) 的布尔函数，可能的最大项有：

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

m_0 = \lgroup A + B \rgroup

\mathrm{m_{0} \: = \: \lgroup A \: + \: B \rgroup}

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

\mathrm{m_{1} \: = \: \lgroup A \: + \: \overline{B} \rgroup}

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

\mathrm{m_{2} \: = \: \lgroup \overline{A} \: + \: B \rgroup}

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

\mathrm{m_{3} \: = \: \lgroup \overline{A} \: + \: \overline{B} \rgroup}

对于 3 个变量 (A、B、C) 的布尔表达式，可能的最大项有：

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

m_0 = \lgroup A + B + C \rgroup

\mathrm{m_{0} \: = \: \lgroup A \: + \: B \: + \: C \rgroup}

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

\mathrm{m_{1} \: = \: \lgroup A \: + \: B \: + \: \overline{C} \rgroup}

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

\mathrm{m_{2} \: = \: \lgroup A \: + \: \overline{B} \: + \: C \rgroup}

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

\mathrm{m_{3} \: = \: \lgroup A \: + \: \overline{B} \: + \: \overline{C} \rgroup}

m4 := (A̅ + B + C)

\mathrm{m_{4} \: = \: \lgroup \overline{A} \: + \: B \: + \: C \rgroup}

m5 := (A̅ + B + C̅)

\mathrm{m_{5} \: = \: \lgroup \overline{A} \: + \: B \: + \: \overline{C} \rgroup}

m6 := (A̅ + B̅ + C)

\mathrm{m_{6} \: = \: \lgroup \overline{A} \: + \: \overline{B} \: + \: C \rgroup}

m7 := (A̅ + B̅ + C̅)

\mathrm{m_{7} \: = \: \lgroup \overline{A} \: + \: \overline{B} \: + \: \overline{C} \rgroup}

这里，从这两个分别在 2 个变量和 3 个变量中的逻辑表达式中，我们可以看到具有两个变量的逻辑函数有四个（22 = 4）最大项，并且具有 3 个变量的逻辑函数有 8 个（23 = 8）最大项。在这种情况下，以非补码形式显示（变量上方有横线）的变量的值等于 0，并且以补码形式显示的变量的值等于 1。

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

这就是布尔代数中最小项和最大项的全部内容。从上面的讨论中，我们可以得出结论，当一个表达式以其标准化的乘积和（SSOP）形式表示时，最小项是它一个逻辑表达式的乘积项。另一方面，当一个逻辑表达式以标准化的和的乘积（SPOS）形式表示时，最大项是该逻辑表达式的和项。

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.

最小项和最大项的共同点是它们都包含逻辑函数的每个“n”个变量。

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