A traditional or crisp set can formally be defined as the following:

A subset U of a set S is a mapping from the elements of S to the elements of the set {0,1}. This is represented by the notation:

U: S-> {0,1}

The mapping is represented by one ordered pair for each element S where the first element is from the set S and the second element is from the set {0,1}. The value zero represents non-membership, while the value one represents membership.

Essentially this says that an element of the set S is either a member or a non-member of the subset U. There are no partial members in traditional sets.

Here is an example of a traditional set:

Consider a set X that contains all the real numbers between 0 and 10 and a subset A of the set X that contains all the real numbers between 5 and 8. Subset A is represented in the figure below.

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In the figure, the interval on the x-axis between 5 and 8 has y-value of one. This indicates that any number in this interval is a member of the subset A. Any number that has a y-value of zero is considered to be a non-member of the subset A.