Changes

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:

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| &nbsp;

<math>\begin{array}{lll}

[| \downharpoonright s \downharpoonleft |]

| <math>[| \downharpoonleft s \downharpoonright |]</math>

& = & [| F |]

| <math>=\!</math>

\\

| <math>[| F |]\!</math>

& = & F^{-1} (\underline{1})

| &nbsp;

| <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>

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