The # Operator

I recall seeing this problem used as a means of testing software that does
Mathematical deductions. I though it might also make for an entertaining
recreational algebra problem.

We have a set U with a binary operation #. It is not known if # is
commutatve, or if # is associative. What is known is that # satifies the
following
axioms:

Axiom 1: For all x, y in U, x # ( x # y ) = x # ( x # x )
Axiom 2: For all x, y, z in U, ( x # y ) # ( y # z ) = y

Give a simple characterization of U and the # operation. Also, if U is a
finite set, what can we say about the cardinality of U?


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