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In mathematics, a finitary relation is defined by one of the formal definitions given below.

  • The basic idea is to generalize the concept of a '' 2-place Relation '', such as the relation of '' Equality '' denoted by the sign "=" in a statement like "5 + 7 = 12" or the relation of '' Order '' denoted by the sign "<" in a statement like "5 < 12". Relations that involve two 'places' or 'roles' are called '' Binary Relation s'' by some and ''dyadic relations'' by others, the latter being historically prior but also useful when necessary to avoid confusion with Binary (base 2) Numerals .


  • The next step up is to consider relations that involve increasing but still finite numbers of places or roles. These are called ''finite place'' or ''finitary'' relations. A finitary relation that involves ''k'' places is variously called a ''k-ary'', a ''k-adic'', or a ''k-dimensional'' relation. The number ''k'' is then called the '' Arity '', the ''adicity'', or the '' Dimension '' of the relation, respectively.



INFORMAL INTRODUCTION


The definition of ''relation'' given in the next Section formally captures a concept that is actually quite familiar from everyday life. For example, consider the relationship, involving three roles that people might play, expressed in a statement of the form "''X'' suspects that ''Y'' likes ''Z'' ". The facts of a concrete situation could be organized in a Table like the following:

Each row of the Table records a fact or makes an assertion of the form "''X'' suspects that ''Y'' likes ''Z'' ". For instance, the first row says, in effect, "Alice suspects that Bob likes Denise". The Table represents a relation ''S'' over the set ''P'' of people under discussion:

: ''P'' = {Alice, Bob, Charles, Denise}.

The data of the Table are equivalent to the following set of ordered triples:

: ''S'' = {(Alice, Bob, Denise), (Charles, Alice, Bob), (Charles, Charles, Alice), (Denise, Denise, Denise)}.

By a slight overuse of notation, it is usual to write ''S''(Alice, Bob, Denise) to say the same thing as the first row of the Table. The relation ''S'' is a ''ternary'' relation, since there are ''three'' items involved in each row. The relation itself is a mathematical object, defined in terms of concepts from Set Theory , that carries all of the information from the Table in one neat package.

The Table for relation ''S'' is an extremely simple example of a Relational Database . The theoretical aspects of databases are the specialty of one branch of Computer Science , while their practical impacts have become all too familiar in our everyday lives. Computer scientists, logicians, and mathematicians, however, tend to see different things when they look at these concrete examples and samples of the more general concept of a relation.

For one thing, databases are designed to deal with empirical data, and experience is always finite, whereas mathematics is nothing if not concerned with infinity, at the very least, potential infinity. This difference in perspective brings up a number of ideas that are usefully introduced at this point, if by no means covered in depth.


EXAMPLE: DIVISIBILITY