Changes

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>

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 <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>

 

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: