Changes

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 <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.

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.

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>\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 multiplication sign <math>^{\backprime\backprime} \cdot ^{\prime\prime}</math> is often omitted from written expressions.

{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:50%"

|+ Table 8.  Product Operation for the Boolean Domain

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"

+ '''Table 8.  Product Operation for the Boolean Domain'''

|- style="background:whitesmoke"

|- style="background:whitesmoke"

| <math>\underline\cdot</math>

| <math>\cdot\!</math>

| <math>\underline{0}</math>

| <math>\underline{0}</math>

| <math>\underline{1}</math>

| <math>\underline{1}</math>

|}

|}

<pre>

<br>

The "sum" of x and y, for values x, y ? B, is presented in Table 9.  Viewed as a function of logical values, + : B?B �> B, this corresponds to the logical operation that is commonly called "exclusive disjunction" and that is otherwise expressed as "x or y, but not both" .  Depending on the context, other signs and readings that invoke this operation are:  "x ? y", read as "x is not equal to y" or as "exactly one of x and y", and "x <?> y", read as "x is not equivalent to y" or as "x opposes y".

 

The ''sum'' of <math>x\!</math> and <math>y,\!</math> for values <math>x, y \in \underline\mathbb{B},</math> is presented in Table&nbsp;9.  Viewed as a function of logical values, <math>+ : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> this corresponds to the logical operation that is commonly called ''exclusive disjunction'' and otherwise read as "<math>x\!</math> or <math>y\!</math> but not both".  Depending on the context, other signs and readings that invoke this operation are:  <math>^{\backprime\backprime} x \ne y ^{\prime\prime},</math> read as "<math>x\!</math> is not equal to <math>y\!</math>" or as "exactly one of <math>x\!</math> and <math>y\!</math> is true", and <math>^{\backprime\backprime} x \not\equiv y ^{\prime\prime},</math> read as "<math>x\!</math> is not equivalent to <math>y\!</math>".

 

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"

|+ '''Table 9. Sum Operation for the Boolean Domain'''

For sentences, the signs of equality ("=") and inequality ("=/=") are reserved to mean the syntactic identity and non�identity, respectively, of their literal strings of characters, while the signs of equivalence ("<=>") and inequivalence ("<=/=>") refer to the logical values, if any, of these strings, and signify the equality and inequality, respectively, of their conceivable boolean values.  For the logical values themselves, the two pairs of symbols collapse in their significance to a single pair, signifying a single form of coincidence or a single form of distinction, respectively, between the boolean values of the entities involved.

For sentences, the signs of equality ("=") and inequality ("=/=") are reserved to mean the syntactic identity and non�identity, respectively, of their literal strings of characters, while the signs of equivalence ("<=>") and inequivalence ("<=/=>") refer to the logical values, if any, of these strings, and signify the equality and inequality, respectively, of their conceivable boolean values.  For the logical values themselves, the two pairs of symbols collapse in their significance to a single pair, signifying a single form of coincidence or a single form of distinction, respectively, between the boolean values of the entities involved.