Changes

For each choice of propositions <math>p\!</math> and <math>q\!</math> about things in <math>X,\!</math> the stretch of <math>F\!</math> to <math>p\!</math> and <math>q\!</math> on <math>X\!</math> is just another proposition about things in <math>X,\!</math> a simple proposition in its own right, no matter how complex its current expression or its present construction as <math>F^\$ (p, q) = \underline{(}~p~,~q~\underline{)}^\$</math> makes it appear in relation to <math>p\!</math> and <math>q.\!</math>  Like any other proposition about things in <math>X,\!</math> it indicates a subset of <math>X,\!</math> namely, the fiber that is variously described in the following ways:

For each choice of propositions <math>p\!</math> and <math>q\!</math> about things in <math>X,\!</math> the stretch of <math>F\!</math> to <math>p\!</math> and <math>q\!</math> on <math>X\!</math> is just another proposition about things in <math>X,\!</math> a simple proposition in its own right, no matter how complex its current expression or its present construction as <math>F^\$ (p, q) = \underline{(}~p~,~q~\underline{)}^\$</math> makes it appear in relation to <math>p\!</math> and <math>q.\!</math>  Like any other proposition about things in <math>X,\!</math> it indicates a subset of <math>X,\!</math> namely, the fiber that is variously described in the following ways:

<pre>

[| F^$ (p, q) |] = [| -(p, q)-^$ |]

 

                  = (F^$ (p, q))^(-1)(%1%)

| <math>[| F^\$ (p, q) |]</math>

 

| <math>=\!</math>

                  = {x in X : F^$ (p, q)(x)}

| <math>[| \underline{(}~p~,~q~\underline{)}^\$ |]</math>

 

                  = {x in X : -(p, q)-^$ (x)}

 

| <math>=\!</math>

                  = {x in X : -(p(x), q(x))- }

| <math>(F^\$ (p, q))^{-1} (\underline{1})</math>

 

                  = {x in X : p(x) ± q(x)}

 

| <math>=\!</math>

                  = {x in X : p(x) =/= q(x)}

| <math>\{~ x \in X ~:~ F^\$ (p, q)(x) ~\}</math>

 

                  = {x in X : -{P}- (x) =/= -{Q}- (x)}

 

| <math>=\!</math>

                  = {x in X : x in P <=/=> x in Q}

| <math>\{~ x \in X ~:~ \underline{(}~p~,~q~\underline{)}^\$ (x) ~\}</math>

 

                  = {x in X : x in P-Q or x in Q-P}

 

| <math>=\!</math>

                  = {x in X : x in P-Q |_| Q-P}

| <math>\{~ x \in X ~:~ \underline{(}~p(x)~,~q(x)~\underline{)} ~\}</math>

 

|-

                  = {x in X : x in P ± Q}

 

| <math>=\!</math>

                  = P ± Q         c  X

| <math>\{~ x \in X ~:~ p(x) + q(x) ~\}</math>

 

                  = [|p|] ± [|q|] c  X.

 

| <math>=\!</math>

| <math>\{~ x \in X ~:~ p(x) \neq q(x) ~\}</math>

| <math>=\!</math>

| <math>\{~ x \in X ~:~ \upharpoonleft P \upharpoonright (x) ~\neq~ \upharpoonleft Q \upharpoonright (x) ~\}</math>

| <math>=\!</math>

| <math>\{~ x \in X ~:~ x \in P ~\nLeftrightarrow~ x \in Q ~\}</math>

| <math>=\!</math>

| <math>\{~ x \in X ~:~ x \in P\!-\!Q ~\operatorname{or}~ x \in Q\!-\!P ~\}</math>

| <math>=\!</math>

| <math>\{~ x \in X ~:~ x \in P\!-\!Q ~\cup~ Q\!-\!P ~\}</math>

| <math>=\!</math>

| <math>\{~ x \in X ~:~ x \in P + Q ~\}</math>

| <math>=\!</math>

| <math>P + Q ~\subseteq~ X</math>

| <math>=\!</math>

| <math>[|p|] + [|q|] ~\subseteq~ X</math>

==References==

==References==