Sets and applications

Sets

In mathematics, we often en counter "sets", for example, real numbers from a set. Defining a set formally is a delicate matter, so we will use "naive" set theory, based on the intuitive properties of sets.

Definition

A set[1] is a collection of objects called elements. We use uppercase letters to label sets, and elements will usually be represented by lower case letters. When is an element of a set , we write , otherwise, we write if contains no elements, it is the empty set, denoted   or . Two sets are equal if they have exactly the same elements. In other words .

Example

The sets and are the same, because the ordering does not matter. The set is also the same set as , because we are not interested in repetitions.

Example

The set is implicit.

Definition

The cardinal of a set is number of distinct elements of . If is finite, the is said to be finite. Otherwise, is said to be infinite.

Example

  1. while .

  2. .

  3. The set of primer numbers is infinite.

  1. 1

    O'LEARY, Michael L. A first course in mathematical logic and set theory. John Wiley & Sons, 2015.

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