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MyWikiBiz, Author Your Legacy — Tuesday November 26, 2024
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===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>
x y z
O
O
</pre>
</pre>
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 — for historical reasons called the "existential interpretation" of logical graphs — 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>
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:
