GeoGebra Classroom

Sets and Subsets: the Basics

What is a Set?

A set is an unordered collection of objects, usually called elements or members of the set. We can represent a set by listing its elements, or by defining the property of all the elements in the set (set-builder notation) Example: A={60, 62, 64, 66, 68, 70} and A={x | 60 ≤ x < 71, x is an even integer} are two representations of the same set. This symbol is used to say that an element belongs to a set, for example . The symbol is used to say that an element does not belong to a set, for example . We can also represent sets geometrically, using Venn diagrams: closed lines enclose portions of the plane that represent the sets, whose elements are represented with points inside the closed lines.

Some "special" sets

The empty set is denoted with the symbol ∅, and is a set that has no elements. The universal set is in general denoted as , and contains all the possible elements from which it's possible to extract the elements of a set. The Venn representation of the universal set is a rectangle.

Subsets

Given two sets and , we say that is a subset of if every element of is also an element of , and we write this as . is a proper subset of if it is not the empty set, and there exists at least one element of that is not in : we write this as .