A Generalization of the Idea of Disjunction

A Generalization of the Idea of Disjunction Kerry M. Soileau July 13, 2010

ABSTRACT We generalize the concept of disjunction.

1. INTRODUCTION

Let X be a nonempty set. Definition 1: We say that a relation  on X is disjunctive if and only if for every 1 2 , x x X  ,   1 2 y X x y x y    if and only if 1 2 x x 

(1) 1 2 2 1 x x x x   

(2)   1 1 x x  

(3)

Examples: (1) Let X be some nonempty set, and define 1 2 x x  if and only if 1 2 x x  . (2) Take X to be the power set of some nonempty set, and define 1 2 x x  if and only if 1 2 x x  . (3) Take   , , X a b c  , and define 1 2 x x  if and only if    1 2 , , x x a b  . Proposition 1: For any 1 2 , x x X  , 1 2 1 2 x x x x    . Proof: Suppose for a contradiction that 1 2 x x  and 1 2 x x  . This means 1 1 x x  , which contradicts (3). ■

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2. A DISJUNCTIVE RELATION INDUCES A PARTIAL ORDER

Definition 2: Define the order satisfying 1 2 x x if and only if   2 1 y X x y x y    . Proposition 2: If 1 2 x x  , then 3 2 x x implies 1 3 x x  . Proof: Suppose 1 2 x x  and 3 2 x x . Since 1 2 x x  , from (2) we get 2 1 x x  . Since 3 2 x x , we have
  2 3 y X x y x y    so in particular 2 1 3 1 x x x x    and thus 3 1 x x  . Another application of (2) yields 1 3 x x  . ■ Proposition 3: is a partial order on X . Proof: (1) Fix x X  . Since clearly   y X x y x y   , it follows that x x . (2) Suppose 1 2 x x
and 2 1 x x . Then   2 1 y X x y x y    and   1 2 y X x y x y    , hence   1 2 y X x y x y    , which by definition of  implies 1 2 x x  . (3) Suppose 1 2 x x and 2 3 x x . Fix y X  . Then 2 1 x y x y   and 3 2 x y x y  , implying 3 1 x y x y  , thus 1 3 x x .■ Example:
Take   , , X a b c  , and define 1 2 x x  if and only if     1 2 , , x x a b  or     1 2 , , x x b a  . Then 1 2 x x if and only if 2x c  . This example can be illustrated with a table:  a b c a F T F b T F F c F F F

The entry for each row and column gives the truth value of the disjunctive relation, for example the table indicates that a b  is true and b c  is false. When a table is constructed for a given disjunctive relation, it will be noticed that the table has three properties:

1. The diagonal contains only entries of F ,
2. the table is symmetric with respect to transposing rows and columns, and
3. no two columns or are alike (and the same is true for rows).

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3. BIBLIOGRAPHY

Bernd S. W. Schröder, Ordered Sets: An Introduction (Boston: Birkhäuser, 2003) 2. George Grätzer (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman and Co. ISBN 0-7167-0442-0 3. Ferreirós, Jose, 2007 (1999). Labyrinth of Thought: A history of set theory and its role in modern mathematics. Basel, Birkhäuser. ISBN 978-3-7643-8349-7 4. Halmos, P.R., Naive Set Theory, D. Van Nostrand Company, Princeton, NJ, 1960. Reprinted, Springer- Verlag, New York, NY, 1974, ISBN 0-387-90092-6.

International Space Station Program Office, Avionics and Software Office, Mail Code OD, NASA Johnson Space Center, Houston, TX 77058 E-mail address: ksoileau@yahoo.com

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