Set

A collection of objects, known as the elements of the set, specified in such a way that we can tell in principle whether or not a given object belongs to it.

E.g. the set of all prime numbers, the set of zeros of the cosine function.

For each set there is a predicate (or property) which is true for (posessed by) exectly those objects which are elements of the set.

The predicate may be defined by the set or vice versa.

Order and repetition of elements within the set are irrelevant so, for example, 1, 2, 3 = 3, 2, 1 = 1, 3, 1, 2, 2.

Some common set of numbers are given the following names:

N = the natural numbers 0, 1, 2, ...

Z = the integers ..., -2, -1, 0, 1, 2, ...

Q = the rational numbers p/q where p, q are in Z and q /= 0.

R = the real numbers

C = the complex numbers.

The empty set is the set with no elements.

The intersection of two sets X and Y is the set containing all the elements x such that x is in X and x is in Y.

The union of two sets is the set containing all the elements x such that x is in X or x is in Y.