Changes

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.

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 following propositional expression:

: (p (q))(p (r)),

| <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))},</math>

in other words,

in other words,

: [p &rArr; q] &and; [p &rArr; r],

| <math>(p \Rightarrow q) \land (p \Rightarrow r),</math>

we are also looking at a picture of:

we are also looking at a picture of:

: (p (q r)),

| <math>\texttt{(} p \texttt{(} q r \texttt{))},</math>

in other words,

in other words,

: p &rArr; [q &and; r].

| <math>p \Rightarrow (q \land r).</math>

Let us now examine the following propositional equation:

Let us now examine the following propositional equation: