8.6.1. Ordinary Set

Let W, a non-empty set, be the formal basis of our further exertions. Set W is often called the universe of discourse or frame of discernment. Our focus is primarily on finite sets. In such cases, the number of elements in W, its cardinality is abbreviated by ‘W‘. Any element in W is denoted by w.

For a specific w Œ W, $ set A which makes either w Œ A or w œ A. This is the basic requirement in ordinary set theory.

Set A is denoted by A = {w1, w2, ... , wn} , wi is the ith element of set A. When elements in A cannot be explicitly listed, A is denoted by { w‘ .... }. The later part in the brackets is a description to those elements which is included. In general,

A = { w‘A(w) true } ,

where A is a function of w.

Given A, B defined on W, if for any w Œ W we have w Œ A Þ w Œ B, then

A B

If A B and B A, then A = B

Any A defined on W is called a subset of W

A W .

An empty set is one that does not contain any element in W. An empty set is denoted as f.

Any A on W , f A W .

A discussed so far is called a single element set. When any A W becomes an element of another set U, U is also a set, it is sometimes called a set class. All the set classes for W becomes 2W . For instance, if W = {black, white} then 2W = {{black, white}, {black}, {white}, f} . In fact, sets defined on W could be a set class. Therefore, a set A defined on W is sometimes denoted as A Π2W .