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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>
 
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>
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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''.
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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>
 
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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 &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'' 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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