Relation mathematics: Encyclopedia II - Relation mathematics - Informal introduction

Relation mathematics - 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:

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

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 conerned 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.

Adapted from the Wikipedia article "Informal introduction", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki