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| The same connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> can also be read as a proposition about things in the universe <math>X = \underline\mathbb{B}^2.</math> If <math>s\!</math> is a sentence that denotes the proposition <math>F,\!</math> then the corresponding assertion says exactly what one states in uttering the sentence <math>^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}.</math> In such a case, one has <math>\downharpoonleft s \downharpoonright \, = F,</math> and all of the following expressions are ordinarily taken as equivalent descriptions of the same set: | | The same connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> can also be read as a proposition about things in the universe <math>X = \underline\mathbb{B}^2.</math> If <math>s\!</math> is a sentence that denotes the proposition <math>F,\!</math> then the corresponding assertion says exactly what one states in uttering the sentence <math>^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}.</math> In such a case, one has <math>\downharpoonleft s \downharpoonright \, = F,</math> and all of the following expressions are ordinarily taken as equivalent descriptions of the same set: |
| | | |
− | {| align="center" cellpadding="8" width="90%" | + | {| align="center" cellpadding="4" style="text-align:left" width="90%" |
− | | | + | | |
− | <math>\begin{array}{lll} | + | |- |
− | [| \downharpoonright s \downharpoonleft |] | + | | <math>[| \downharpoonleft s \downharpoonright |]</math> |
− | & = & [| F |]
| + | | <math>=\!</math> |
− | \\
| + | | <math>[| F |]\!</math> |
− | \\
| + | |- |
− | & = & F^{-1} (\underline{1}) | + | | |
− | \\
| + | | <math>=\!</math> |
− | \\ | + | | <math>F^{-1} (\underline{1})</math> |
− | & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ s ~\}
| + | |- |
− | \\
| + | | |
− | \\ | + | | <math>=\!</math> |
− | & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ F(x, y) = \underline{1} ~\}
| + | | <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ s ~\}</math> |
− | \\
| + | |- |
− | \\ | + | | |
− | & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ F(x, y) ~\}
| + | | <math>=\!</math> |
− | \\
| + | | <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ F(x, y) = \underline{1} ~\}</math> |
− | \\ | + | |- |
− | & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \underline{(}~x~,~y~\underline{)} = \underline{1} ~\}
| + | | |
− | \\
| + | | <math>=\!</math> |
− | \\ | + | | <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ F(x, y) ~\}</math> |
− | & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \underline{(}~x~,~y~\underline{)} ~\}
| + | |- |
− | \\
| + | | |
− | \\ | + | | <math>=\!</math> |
− | & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x ~\operatorname{exclusive~or}~ y ~\}
| + | | <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \underline{(}~x~,~y~\underline{)} = \underline{1} ~\}</math> |
− | \\
| + | |- |
− | \\ | + | | |
− | & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \operatorname{just~one~true~of}~ x, y ~\}
| + | | <math>=\!</math> |
− | \\
| + | | <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \underline{(}~x~,~y~\underline{)} ~\}</math> |
− | \\ | + | |- |
− | & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x ~\operatorname{not~equal~to}~ y ~\}
| + | | |
− | \\
| + | | <math>=\!</math> |
− | \\ | + | | <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x ~\operatorname{exclusive~or}~ y ~\}</math> |
− | & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x \nLeftrightarrow y ~\}
| + | |- |
− | \\
| + | | |
− | \\ | + | | <math>=\!</math> |
− | & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x \neq y ~\}
| + | | <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \operatorname{just~one~true~of}~ x, y ~\}</math> |
− | \\
| + | |- |
− | \\ | + | | |
− | & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x + y ~\}.
| + | | <math>=\!</math> |
− | \end{array}</math>
| + | | <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x ~\operatorname{not~equal~to}~ y ~\}</math> |
| + | |- |
| + | | |
| + | | <math>=\!</math> |
| + | | <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x \nLeftrightarrow y ~\}</math> |
| + | |- |
| + | | |
| + | | <math>=\!</math> |
| + | | <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x \neq y ~\}</math> |
| + | |- |
| + | | |
| + | | <math>=\!</math> |
| + | | <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x + y ~\}.</math> |
| + | |- |
| + | | |
| |} | | |} |
| | | |