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| − | ==Background Material==
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| − | ===Propositions and Sentences===
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| − | * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems_:_Part_2#2.2.3._Propositions_and_Sentences Inquiry Driven Systems • Propositions and Sentences]
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| − | 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.
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| − | 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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| − | <br>
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| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
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| − | |+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Product Operation for the Boolean Domain}\!</math>
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| − | |- style="height:40px; background:ghostwhite"
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| − | | <math>\cdot\!</math>
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| − | | <math>\underline{0}</math>
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| − | | <math>\underline{1}</math>
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| − | |-
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| − | | style="background:ghostwhite" | <math>\underline{0}</math>
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| − | | <math>\underline{0}</math>
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| − | | <math>\underline{0}</math>
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| − | | style="background:ghostwhite" | <math>\underline{1}</math>
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| − | | <math>\underline{0}</math>
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| − | | <math>\underline{1}</math>
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| − | |}
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| − | <br>
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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\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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| − | <br>
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| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
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| − | |+ style="height:30px" | <math>\text{Table 10.} ~~ \text{Sum Operation for the Boolean Domain}\!</math>
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| − | |- style="height:40px; background:ghostwhite"
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| − | | <math>+\!</math>
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| − | | <math>\underline{0}</math>
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| − | | <math>\underline{1}</math>
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| − | |-
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| − | | style="background:ghostwhite" | <math>\underline{0}</math>
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| − | | <math>\underline{0}</math>
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| − | | <math>\underline{1}</math>
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| − | |-
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| − | | style="background:ghostwhite" | <math>\underline{1}</math>
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| − | | <math>\underline{1}</math>
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| − | | <math>\underline{0}</math>
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| − | |}
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| − | <br>
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| − | ===Logical Implication===
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| − | * [http://intersci.ss.uci.edu/wiki/index.php/Logical_implication Logical Implication]
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| − | The concept of logical implication is associated with an operation on two logical values, typically the values of two propositions, that produces a value of ''false'' just in case the first operand is true and the second operand is false.
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| − | In the interpretation where <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}</math>, the truth table associated with the statement <math>{}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime},</math> symbolized as <math>{}^{\backprime\backprime} p \Rightarrow q {}^{\prime\prime},</math> appears below:
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| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
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| − | |+ style="height:30px" | <math>\text{Logical Implication}\!</math>
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| − | |- style="height:40px; background:#f0f0ff"
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| − | | style="width:33%" | <math>p\!</math>
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| − | | style="width:33%" | <math>q\!</math>
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| − | | style="width:33%" | <math>p \Rightarrow q\!</math>
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| − | | <math>0\!</math> || <math>0\!</math> || <math>1\!</math>
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| − | |-
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| − | | <math>0\!</math> || <math>1\!</math> || <math>1\!</math>
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| − | |-
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| − | | <math>1\!</math> || <math>0\!</math> || <math>0\!</math>
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| − | |-
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| − | | <math>1\!</math> || <math>1\!</math> || <math>1\!</math>
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| − | |}
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| − | <br>
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| − | ===Work Area===
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| − | <br>
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| − | {| align="center" cellspacing="6"
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| − | <math>\begin{array}{|c||cc|}
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| − | \hline
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| − | \texttt{=}\!\texttt{<} & 0 & 1 \\
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| − | \hline\hline
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| − | 0 & 1 & 1 \\
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| − | 1 & 0 & 1 \\
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| − | \hline
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| − | \end{array}\!</math>
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| − | |}
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| − | <br>
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| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
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| − | |- style="height:40px; background:ghostwhite"
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| − | | style="width:33%" | <math>\texttt{=}\!\texttt{<}\!</math>
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| − | | <math>0\!</math>
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| − | | <math>1\!</math>
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| − | | style="background:ghostwhite; width:33%" | <math>0\!</math>
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| − | | <math>1\!</math>
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| − | | <math>1\!</math>
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| − | | style="background:ghostwhite; width:33%" | <math>1\!</math>
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| − | | <math>0\!</math>
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| − | | <math>1\!</math>
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| − | |}
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| − | <br>
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| | ==Commentary Work Area== | | ==Commentary Work Area== |
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