# Describing and Defining Sets

This lesson is part of a series of lessons on

sets.

In this lesson, we will learn

- how to define sets and set notations
- subsets and proper subsets
- Venn diagrams and set operations

Related Topics:

More Lessons on Sets
A

**set** is a collection of objects, things or symbols which are

clearly defined.

The individual objects in a set are called the

members or

elements of the set.

A set must be properly defined so that we can find out whether an object is a member of the set.

**1. Listing the elements **(Listing Method)

The set can be defined by listing all its elements, separated by commas and enclosed within braces. This is called the roster method.

*Example**:*

*B = * {2, 4, 6, 8, 10}

*X* = {*a, b, c, d, e*}

However, in some instances, it may not be possible to list all the elements of a set. In such cases, we could define the set by method 2.

**2. Describing the elements** (Roster Method)

The set can be defined, where possible, by describing the elements.
This is called the set-builder notation.

*Example:*

*C* = {*x* : *x *is an integer, *x > – *3 }

This is read as: “*C* is the set of elements *x* such that *x* is an integer greater than –3.”

*D*= {*x*: *x* is a river in a state}

We should describe a certain property which all the elements *x*, in a set, have in common so that we can know whether a particular thing belongs to the set.

We relate a member and a set using the symbol ∈. If an object *x* is an element of set *A*, we write* x* ∈ *A*. If an object *z* is not an element of set *A*, we write *z* ∉ *A*.

∈ denotes “is an element of’ or “is a member of” or “belongs to”

∉ denotes “is not an element of” or “is not a member of” or “does not belong to”

Example:

If *A* = {1, 3, 5} then 1 ∈ *A * and 2 ∉ *A*

This video introduces the basic vocabulary used in set theory.

This video defines and give the notation used for subsets and proper subsets.

This video introduces Venn diagrams and set operations.

This video provides examples of basic Venn diagrams and set operations.