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Directory:Jon Awbrey/Papers/Inquiry Driven Systems (view source)
Revision as of 19:56, 14 January 2009
, 19:56, 14 January 2009→1.3.10.3. Propositions and Sentences: fold in changes from later version
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. 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. 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'' <math>x \cdot y</math> of two values <math>x\!</math> and <math>y\!</math> in <math>\underline\mathbb{B}</math> is given by Table 9. As a dyadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> the product corresponds to the logical operation of ''conjunction'', written <math>^{\backprime\backprime} x \land y ^{\prime\prime}</math> or <math>^{\backprime\backprime} x\!\And\!y ^{\prime\prime}</math> and read as <math>^{\backprime\backprime} x ~\operatorname{and}~ y ^{\prime\prime}.</math> In accord with common practice, the multiplication sign is often omitted from written expressions.
The ''product'' <math>x \cdot y</math> of two values <math>x\!</math> and <math>y\!</math> in <math>\underline\mathbb{B}</math> is given by Table 9. As a dyadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> the product corresponds to the logical operation of ''conjunction'', written <math>^{\backprime\backprime} x \land y ^{\prime\prime}</math> or <math>^{\backprime\backprime} x\!\And\!y ^{\prime\prime}</math> and read as <math>^{\backprime\backprime} x ~\operatorname{and}~ y ^{\prime\prime}.</math> In accord with common practice, the multiplication sign is frequently omitted from written expressions of the product.
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The ''sum'' <math>x + y\!</math> of two values <math>x\!</math> and <math>y\!</math> in <math>\underline\mathbb{B}</math> is given in Table 10. As a dyadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> the sum corresponds to the logical operation of ''exclusive disjunction'', usually read as <math>^{\backprime\backprime} x ~\text{or}~ y ~\text{but not both} ^{\prime\prime}.</math> Depending on the context, other signs and readings that invoke this operation are: <math>^{\backprime\backprime} x \ne y ^{\prime\prime}</math> or <math>^{\backprime\backprime} x \not\equiv y ^{\prime\prime},</math> read as <math>^{\backprime\backprime} x ~\text{is not equal to}~ y ^{\prime\prime},</math> <math>^{\backprime\backprime} x ~\text{is not equivalent to}~ y ^{\prime\prime},</math> or <math>^{\backprime\backprime} \text{exactly one of}~ x, y ~\text{is true} ^{\prime\prime}.</math>
The ''sum'' <math>x + y\!</math> of two values <math>x\!</math> and <math>y\!</math> in <math>\underline\mathbb{B}</math> is given in Table 10. As a dyadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> the sum corresponds to the logical operation of ''exclusive disjunction'', usually read as <math>^{\backprime\backprime} x ~\text{or}~ y ~\text{but not both} ^{\prime\prime}.</math> Depending on the context, other signs and readings that invoke this operation are: <math>^{\backprime\backprime} x \ne y ^{\prime\prime}</math> or <math>^{\backprime\backprime} x \not\Leftrightarrow y ^{\prime\prime},</math> read as <math>^{\backprime\backprime} x ~\text{is not equal to}~ y ^{\prime\prime},</math> <math>^{\backprime\backprime} x ~\text{is not equivalent to}~ y ^{\prime\prime},</math> or <math>^{\backprime\backprime} \text{exactly one of}~ x, y ~\text{is true} ^{\prime\prime}.</math>
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