Thinking about MINUS

From: Walt <wamitty_at_verizon.net>
Date: Sat, 06 Jan 2007 20:29:07 GMT
Message-ID: <nETnh.1011$us1.760@trndny04>

The discussion over in "Curious SQL question" started me thinking (after I got over my embarassment at posting a wrong solution).

Suppose you started with these two primitive concepts:

A universal set, called U (whatever that is)

and

MINUS(A,B), a function that removes from set A any elements common to A and B.

Can you derive the rest of it? Here's my first attempt:

Infix notation: A MINUS B = MINUS (A,B). Just a notational convenience for me. This should be trivial. I apologize to any readers who find this inconvenient.

Empty Set: PHI = U MINUS U. Note that PHI is somehow "bound" to U. Whether the PHIs of different universes are or are not the same PHI is something I'll let the rest of you discuss.

NOT operator. NOT(A) = U MINUS A.

Left association. (A MINUS B) MINUS C = (A MINUS C) MINUS B (proof omitted)

INTERSECTION. A INTERSECT B = A MINUS (A MINUS B)

• B MINUS (B MINUS A)

UNION. A UNION B = NOT (NOT (A) INTERSECT NOT (B)) From here, it looks like we can bootstrap our way up to the rest of set theory and Boolean algebra.
Or am I seeing something wrong (again)? Received on Sat Jan 06 2007 - 14:29:07 CST