Changes

===Solution===

===Solution===

[http://mathforum.org/kb/plaintext.jspa?messageID=6514666 Solution posted by Jon Awbrey, working in the medium of logical graphs].

[http://mathforum.org/kb/plaintext.jspa?messageID=6514666 Solution posted by Jon Awbrey, proceeding by way of logical graphs].

In logical graphs, the required equivalence looks like this:

In logical graphs, the required equivalence looks like this:

<pre>

<pre>

           O

           O

</pre>

</pre>

Etc.

And so on.

In this form of representation, for historical reasons called the "existential interpretation" of logical graphs, we have the following expressions for basic logical operations:

In this form of representation &mdash; for historical reasons called the "existential interpretation" of logical graphs &mdash; we have the following expressions of basic logical operations:

The disjunction <math>x \lor y</math> is written <math>((x)(y)).\!</math>

The disjunction <math>x \lor y</math> is written <math>((x)(y)).\!</math>

</pre>

</pre>

Etc.

And so on.

The implication <math>x \Rightarrow y</math> is written <math>(x (y)),\!</math> which can be read "not <math>x\!</math> without <math>y\!</math>" if that helps to remember the form of expression.

The implication <math>x \Rightarrow y</math> is written <math>(x (y)),\!</math> which can be read "not <math>x\!</math> without <math>y\!</math>" if that helps to remember the form of expression.

</pre>

</pre>

Thus, the equivalence <math>x \Leftrightarrow y</math> has to be written somewhat inefficiently as a conjunction of two implications:  <math>(x (y)) (y (x)).\!</math>

Thus, the equivalence <math>x \Leftrightarrow y</math> has to be written somewhat inefficiently as a conjunction of to and fro implications:  <math>(x (y)) (y (x)).\!</math>

This corresponds to the logical graph:

This corresponds to the logical graph: