Sets

Concepts and Notations

In mathematics, the words "collection", "class", or "set" are interchangeable. However, the word "set" is most commonly used.

A "set" is a gathering or collection of distinct objects, which are called elements (or members) of the set.
The number of elements can be finite, infinite, or even none.
Sets are usually designated by upper-case letters and elements by lower-case letters .
A simple way to specify a set is to list its elements (when it is possible). To this end, we put the elements inside a pair of braces (curly brackets) . For example is a set whose elements are 1, and 0.
Sometimes when writing all the elements of a set is not feasible, we may skip some of the elements and replace them by ellipsis. For example, to show the set of positive integers from 1 to 50, we can write .
The ellipsis is used only when the elements follow a pattern that can be clearly understood from the first few elements.
If there is no elements after the ellipsis, it is assumed that the pattern continues forever. For example the set written as contains all positive integers. To show that the elements of a set go on forever in both directions, we may use the ellipsis at both ends. For example, the set of all integers (positive, negative and zero) may be displayed as
Changing the order of the elements or repeating the elements does NOT change the set. Therefore the set is the same as .
When the object is an element of the set , we writeAlternatively we can say " is in ", " belongs to ", or " contains ". When is not an element of , we writeFor example, let . Here is a set whose elements are 1, 2, and 3. In this example, but .

Subsets

We say a set is a subset of a set , and we write

if every element of also belongs to . In other words, for every

We can also say " contains " or " is contained in ."

For example, if , , and then

because every element of is also an element of , but is not a subset of

because but .

Equality

Two sets and are called "equal" or "identical", and we write , if they consist of exactly the same elements.

If two sets and are equal then and . If and , then . That is,

Universal Set

A set that contains all the elements that we want to consider for now is called the "universal set", usually denoted by (or in some books by ). For example, we might say that the universal set is the set of all real numbers or the set of all integers.

The universal set is also called the domain of discourse or the universe of discourse. The universal set may vary from one application to another.

In elementary calculus, the universal set is assumed to be the set of all real numbers unless otherwise stated.

Set Builder Notation

When listing of all elements of a set is not possible, we can specify the set by describing a property common to the elements in the set and only to those elements. For example:

set of all numbers which are positive odd integers

set of all real numbers for which .

When we want to describe a set in this way, we can use set-builder notation. The general form of set-builder notation is

that designates the set of all elements in for which the property is satisfied. The vertical bar which can also be written as a colon ":" is a separator that is used in place of "such that."

In calculus the universal set when not specified is assumed to be the set of real numbers, so we may omit the reference to and simply write

For example,

means the set of all real numbers which are greater than 1 but less than 2.

means the set of all numbers for which .

In the above examples, is a "variable". A variable is a symbol that represents any element of a given set.

Because it does not matter which symbol we use to represent the elements of a set, the letter is a dummy and can be replaced by any other symbol. Thus we may write

Empty Set or Void Set

A set is empty if it has no elements. The empty set or void set is denoted by

Note that a box that contains only an empty box is not empty. Similarly {{ }} is not an empty set, because it has one element: { }.
For every set :

Venn diagrams

Venn diagrams, introduced by the English mathematician John Venn, are useful for visualizing sets and relations between them. In a Venn diagram, we represent a set as a region, often a disk, in the plane and its elements as points. The universal set, conventionally represented by a rectangle, is the outmost shape. If two regions overlap, it means that the two corresponding sets have some elements in common. The following Venn diagrams illustrate the relations we have discussed so far.