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For the sake of simplicity in discussing this example, let's stick with the existential interpretation (<math>\operatorname{Ex}</math>) of logical graphs and their corresponding parse strings.  Under <math>\operatorname{Ex}</math> the formal expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> translates into the vernacular expression <math>{}^{\backprime\backprime} p ~\operatorname{implies}~ q ~\operatorname{and}~ p ~\operatorname{implies}~ r {}^{\prime\prime},</math> in symbols, <math>(p \Rightarrow q) \land (p \Rightarrow r),</math> so this is the reading that we'll want to keep in mind for the present.

For the sake of simplicity in discussing this example, let's stick with the existential interpretation (<math>\operatorname{Ex}</math>) of logical graphs and their corresponding parse strings.  Under <math>\operatorname{Ex}</math> the formal expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> translates into the vernacular expression <math>{}^{\backprime\backprime} p ~\operatorname{implies}~ q ~\operatorname{and}~ p ~\operatorname{implies}~ r {}^{\prime\prime},</math> in symbols, <math>(p \Rightarrow q) \land (p \Rightarrow r),</math> so this is the reading that we'll want to keep in mind for the present. Where brevity is required, we may refer to the propositional expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> under the name <math>f\!</math> by making use of the following definition:

 

Where brevity is required, we may refer to the propositional expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> under the name <math>f\!</math> by making use of the following definition:

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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 a couple of 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 (<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.  (Note.  In this Ascii version, I use [&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;] for 0 and [&nbsp;`&nbsp;` &nbsp;`&nbsp;] for 1.)

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.  (Note.  In this Ascii version, I use [&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;] for 0 and [&nbsp;`&nbsp;` &nbsp;`&nbsp;] for 1.)

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There are a number of standard ways in mathematics and statistics for talking about "the subset ''W'' of the domain ''X'' that gets painted with the value ''z'' by the indicator function ''f'' : ''X'' &rarr; '''B'''".  The subset ''W'' &sube; ''X'' is called by a variety of names in different settings, for example, the ''antecedent'', the ''fiber'', the ''inverse image'', the ''level set'', or the ''pre-image'' in ''X'' of ''z'' under ''f''.  It is notated and defined as ''W'' = ''f''^–1(''z'').  Here, ''f''^–1 is called the ''converse relation'' or the ''inverse relation'' &mdash; it is not in general an inverse function &mdash; corresponding to the function ''f''.  Whenever possible in simple examples, we use lower case letters for functions ''f'' : ''X'' &rarr; '''B''', and its is sometimes useful to employ capital letters for subsets of ''X'', if possible, in such a way that ''F'' is the fiber of 1 under ''f'', in other words, ''F'' = ''f''^–1(1).

There are a number of standard ways in mathematics and statistics for talking about the subset <math>W\!</math> of the functional domain <math>X\!</math> that gets painted with the value <math>z \in \mathbb{B}</math> by the indicator function <math>f : X \to \mathbb{B}.</math> The region <math>W \subseteq X</math> is called by a variety of names in different settings, for example, the ''antecedent'', the ''fiber'', the ''inverse image'', the ''level set'', or the ''pre-image'' in <math>X\!</math> of <math>z\!</math> under <math>f.\!</math> It is notated and defined as <math>W = f^{-1}(z).\!</math> Here, <math>f^{-1}\!</math> is called the ''converse relation'' or the ''inverse relation'' &mdash; it is not in general an inverse function &mdash; corresponding to the function <math>f.\!</math> Whenever possible in simple examples, we use lower case letters for functions <math>f : X \to \mathbb{B},</math> and it is sometimes useful to employ capital letters for subsets of <math>X,\!</math> if possible, in such a way that <math>F\!</math> is the fiber of 1 under <math>f,\!</math> in other words, <math>F = f^{-1}(1).\!</math>

The easiest way to see the sense of the venn diagram is to notice that the expression "(p (q))", read as "''p'' &rArr; ''q''", can also be read as "not ''p'' without ''q''".  Its assertion effectively excludes any tincture of truth from the region of ''P'' that lies outside the region ''Q''.

The easiest way to see the sense of the venn diagram is to notice that the expression "(p (q))", read as "''p'' &rArr; ''q''", can also be read as "not ''p'' without ''q''".  Its assertion effectively excludes any tincture of truth from the region of ''P'' that lies outside the region ''Q''.

Likewise for the expression "(p (r))", read as "''p'' &rArr; ''r''", and also readable as "not ''p'' without ''r''".  Asserting it effectively excludes any tincture of truth from the region of ''P'' that lies outside the region ''R''.

Likewise for the expression "(p (r))", read as "''p'' &rArr; ''r''", and also readable as "not ''p'' without ''r''".  Asserting it effectively excludes any tincture of truth from the region of ''P'' that lies outside the region ''R''.

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 — others may elect to give it the more dignified title of a ''logical quotient topology'' or some such thing — 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'' 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:

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