Changes

Since the expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> involves just three variables, it may be worth the trouble to draw a venn diagram of the situation.  There are in fact two different ways to execute the picture.

Since the expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> involves just three variables, it may be worth the trouble to draw a venn diagram of the situation.  There are in fact two different ways to execute the picture.

Figure&nbsp;1 indicates the points of the universe of discourse <math>X\!</math> for which the proposition <math>f : X \to \mathbb{B}</math> has the value 1, here interpreted as the logical value <math>\operatorname{true}.</math>  In this ''paint by numbers'' style of picture, one simply paints over the cells of a generic template for the universe <math>X,\!</math> going according to some previously adopted convention, for instance:  Let the cells that get the value 0 under <math>f\!</math> remain untinted and let the cells that get the value 1 under <math>f\!</math> be painted or shaded.  In doing this, it may be good to remind ourselves that the value of the picture as a whole is not in the ''paints'', in other words, the <math>0, 1\!</math> in <math>\mathbb{B},</math> but in the pattern of regions that they indicate.

Figure&nbsp;27 indicates the points of the universe of discourse <math>X\!</math> for which the proposition <math>f : X \to \mathbb{B}</math> has the value 1, here interpreted as the logical value <math>\operatorname{true}.</math>  In this ''paint by numbers'' style of picture, one simply paints over the cells of a generic template for the universe <math>X,\!</math> going according to some previously adopted convention, for instance:  Let the cells that get the value 0 under <math>f\!</math> remain untinted and let the cells that get the value 1 under <math>f\!</math> be painted or shaded.  In doing this, it may be good to remind ourselves that the value of the picture as a whole is not in the ''paints'', in other words, the <math>0, 1\!</math> in <math>\mathbb{B},</math> but in the pattern of regions that they indicate.

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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'' &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:

Figure&nbsp;28 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:

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| [[Image:Venn Diagram (P (Q R)).jpg|500px]] || (28)

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| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}</math>

|        /                / P \                \        |

|      |        Q       |      |      R         |       |

|      |                |      |                |      |

|       |                |      |                |      |

|       o                o      o                o      |

|        \                 \    /                /        |

Venn Diagram for (p (q r))

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