Changes

The easiest way to see the sense of the venn diagram is to notice that the expression <math>\texttt{(} p \texttt{(} q \texttt{))},</math> read as <math>p \Rightarrow q,</math> can also be read as <math>{}^{\backprime\backprime} \operatorname{not}~ p ~\operatorname{without}~ q {}^{\prime\prime}.</math>  Its assertion effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>Q.\!</math>  In a similar manner, the expression <math>\texttt{(} p \texttt{(} r \texttt{))},</math> read as <math>p \Rightarrow r,</math> can also be read as <math>{}^{\backprime\backprime} \operatorname{not}~ p ~\operatorname{without}~ r {}^{\prime\prime}.</math>  Asserting it effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>R.\!</math>

The easiest way to see the sense of the venn diagram is to notice that the expression <math>\texttt{(} p \texttt{(} q \texttt{))},</math> read as <math>p \Rightarrow q,</math> can also be read as <math>{}^{\backprime\backprime} \operatorname{not}~ p ~\operatorname{without}~ q {}^{\prime\prime}.</math>  Its assertion effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>Q.\!</math>  In a similar manner, the expression <math>\texttt{(} p \texttt{(} r \texttt{))},</math> read as <math>p \Rightarrow r,</math> can also be read as <math>{}^{\backprime\backprime} \operatorname{not}~ p ~\operatorname{without}~ r {}^{\prime\prime}.</math>  Asserting it effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>R.\!</math>

Figure&nbsp;2 shows the other standard way of drawing a venn diagram for such a proposition.  In this ''punctured soap film'' style of picture &mdash; others may elect to give it the more dignified title of a ''logical quotient topology'' or some such thing &mdash; one goes on from the previous picture to collapse the fiber of 0 under ''X'' down to the point of vanishing utterly from the realm of active contemplation, thereby arriving at a degenre of picture like so:

Figure&nbsp;2 shows the other standard way of drawing a venn diagram for such a proposition.  In this ''punctured soap film'' style of picture &mdash; others may elect to give it the more dignified title of a ''logical quotient topology'' &mdash; one begins with Figure&nbsp;1 and then proceeds to collapse the fiber of 0 under <math>X\!</math> down to the point of vanishing utterly from the realm of active contemplation, arriving at the following picture:

{| align="center" cellpadding="10" style="text-align:center; width:90%"

{| align="center" cellpadding="10" style="text-align:center; width:90%"

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This diagram indicates that the region where ''p'' is true is wholly contained in the region where both ''q'' and ''r'' are true.  Since only the regions that are painted true in the previous figure show up at all in this one, it is no longer necessary to distinguish the fiber of 1 under ''f'' by means of any stipple.

This diagram indicates that the region where <math>p\!</math> is true is wholly contained in the region where both <math>q\!</math> and <math>r\!</math> are true.  Since only the regions that are painted true in the previous figure show up at all in this one, it is no longer necessary to distinguish the fiber of 1 under <math>f\!</math> by means of any shading.

In sum, it is immediately obvious from the venn diagram that in drawing a representation of the propositional expression:

In sum, it is immediately obvious from the venn diagram that in drawing a representation of the propositional expression: