Changes

The ''negation'' of <math>x,\!</math> for <math>x \in \underline\mathbb{B},</math> written <math>^{\backprime\backprime} \underline{(} x \underline{)} ^{\prime\prime}</math> and read <math>^{\backprime\backprime} \operatorname{not}\ x ^{\prime\prime},</math> is the boolean value <math>\underline{(} x \underline{)} \in \underline\mathbb{B}</math> that is <math>\underline{1}</math> when <math>x\!</math> is <math>\underline{0},</math> and <math>\underline{0}</math> when <math>x\!</math> is <math>\underline{1}.</math>  In other words, negation is a monadic operation on boolean values, or a function of the form <math>\underline{(} \cdot \underline{)} : \underline\mathbb{B} \to \underline\mathbb{B}.</math>

The ''negation'' of <math>x,\!</math> for <math>x \in \underline\mathbb{B},</math> written <math>^{\backprime\backprime} \underline{(} x \underline{)} ^{\prime\prime}</math> and read <math>^{\backprime\backprime} \operatorname{not}\ x ^{\prime\prime},</math> is the boolean value <math>\underline{(} x \underline{)} \in \underline\mathbb{B}</math> that is <math>\underline{1}</math> when <math>x\!</math> is <math>\underline{0},</math> and <math>\underline{0}</math> when <math>x\!</math> is <math>\underline{1}.</math>  In other words, negation is a monadic operation on boolean values, or a function of the form <math>\underline{(} \cdot \underline{)} : \underline\mathbb{B} \to \underline\mathbb{B}.</math>

It is convenient to transport the product and the sum operations of <math>\mathbb{B}</math> into the logical setting of <math>\underline\mathbb{B},</math> where they can be symbolized by signs of the same character, underlined as necessary to avoid confusion.  This yields the following definitions of a ''product'' and a ''sum'' in <math>\underline\mathbb{B}</math> and leads to the following forms of multiplication and addition tables.

It is convenient to transport the product and the sum operations of B into the logical setting of B, where they can be symbolized by signs of the same character, doubly underlined as necessary to avoid confusion.  This yields the following definitions of a "product" and a "sum" in B and leads to the following forms of multiplication and addition tables.

The "product" of x and y, for values x, y ? B, is given by Table 8.  Viewed as a function of logical values, . : B?B �> B, this corresponds to the logical operation that is commonly called "conjunction" and that is otherwise expressed as "x and y".  In accord with common practice, the raised dot ".", doubly underlined or not, is often omitted from written expressions.

The ''product'' of <math>x\!</math> and <math>y,\!</math> for values <math>x, y \in \underline\mathbb{B},</math> is given by Table&nbsp;8.  Viewed as a function of logical values, <math>\underline\cdot : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> this corresponds to the logical operation that is commonly called ''conjunction'' and otherwise expressed as <math>^{\backprime\backprime} x ~\operatorname{and}~ y ^{\prime\prime}.</math> In accord with common practice, the raised dot <math>^{\backprime\backprime} \underline\cdot ^{\prime\prime},</math> underlined or not, is often omitted from written expressions.

Table 8.  Product Operation for the Boolean Domain

Table 8.  Product Operation for the Boolean Domain

. 0 1

. 0 1