Sets

A set is a collection of objects. These objects, called the members or elements of the set, are enclosed in a special type of bracket, called braces {...} and are separated by commas. The order of the elements in not important.

Notation

A set can be described in one of three ways:

1. In words: A = {the first three whole numbers}

2. By listing each member: A = {0, 1, 2}

3. In set builder notation, which states the rules of membership.

A = {x: x < 3, x Y10_Sets_01.gif W}

Symbols

Symbol
Meaning
Example
U or E
 the universal set the set of all elements or members being considered.
Ø or {}
 the empty or null set This set has no elements or members.
Y10_Sets_01.gif
 is a member of 2 Y10_Sets_01.gif  {even numbers}
Y10_Sets_02.gif
 is not a member of 3 Y10_Sets_02.gif {even numbers}
Y10_Sets_03.gif
 is a subset of
All of the sets that can be made using elements of the set. Includes the empty set and the original set.

{3, 4} Y10_Sets_03.gif {2, 3, 4}
The set {2, 3, 4} has 8 subsets.
These are {}, {2}, {3}, {4}, {2, 3}, {3, 4}, {2, 4}, {2, 3, 4}
n(A)

the cardinal number (number of members) of set A. 
The number of elements in a set.

if A = {4, 5, 6, 7, 8}
n(A) = 5
A'
 the complement of set A. 
All the members not in set A.		 If A = {4, 5, 6, 7, 8} and the universal set is {whole numbers less than 10} then 
A' = {0, 1, 2, 3, 9}
Y10_Sets_04.gif
 the union of sets.
Every element from EACH set.

If A = {4, 5, 6, 7, 8} and B = {3, 4, 5} then
A Y10_Sets_04.gif B = {3, 4, 5, 6, 7, 8}
∩

the intersection of sets.
All of the elements COMMON to both sets.

 If A = {4, 5, 6, 7, 8} and B = {3, 4, 5} then
A ∩ B = {4, 5}

 

Types of Sets

Name
Description
Examples
Equal sets A = B if A and B have the same elements. If P = {0, 1, 2} and Q = {2, 1, 0} then P = Q
Equivalent sets A ↔ B if A and B have the same number of elements. If R = {0, 1, 2} and S = {-5, -4, -3} then R↔ S
Disjoint sets Sets that have no elements in common. If A = {10, 11, 12} and B = {3, 4} then A and B are disjoint. 
A ∩ B = Ø
Finite sets A finite set is one that has a countable number of elements. A = {10, 11, 12} is a finite set.
Infinite sets An infinite set has an uncountably large number of elements. W = {0, 1, 2, 3,...} is an infinite set.
Cross or Cartesian product A X B means the set of ordered pairs that is formed by pairing each member of the set A with each member of the set B. If P = {0, 1, 2} and B = {3, 4} 
then P X B = {(0,3), (0, 4), (1,3), (1, 4), (2,3), (2, 4)}

Venn Diagrams

Venn diagrams are useful for showing sets, solving problems and situations involving probability. These are covered in detail in Year 12, Topic 59.