Sets and Other Basic Concepts
Chapter 1, Section 2
 Variables
 A variable is a letter used to represent many numbers.

 x, y, and z are usually used for variables.
 Sometimes letters are also used to represent fixed constants (numbers that do not change).

 a, b, c, and letters other than x, y, z are usually used for fixed constants.
 In the formula ax = b, x is the variable while a and b are fixed constants.
 Sets
 A set is a collection of objects.
 The objects are called elements or members.
 The elements can be anything.
 { 1, 2, 3, 4, 5 } is a set of numbers.
 { dog, cat, mouse, dolphin } is a set of animals.
 Sets are often assigned a capital letter for easy reference.
 Examples:
 A = { 2, 4, 6, 8, ... }
 D = { ..., 4, 2, 0, 2, 4, ... }
 Set Symbols
 In roster form, the elements (or members) of a set are
listed between braces: { ...elements... }
 means "is an element of".
 means "is not an element of".
 Examples:
 2 { 2, 4, 6, 8, ... }
 1 { ..., 4, 2, 0, 2, 4, ... }
 Ø or { } means the empty set or null set, which is a set without elements.
 means "is a subset of".
 means "is not a subset of".
 Subsets
 A set, B, is a subset set of a set, C, if all the elements in B are also in C.
 B C is read "B is a subset of C."
 A set, B, is not a subset set of a set, C, if one of the elements in B is not in C.
 B C is read "B is not a subset of C."
 Sets of Numbers

Set of Numbers  Symbol  Elements 
Natural or Counting  N  { 1, 2, 3, 4, ... } 
Whole  W  { 0, 1, 2, 3, 4, ... } 
Integer  I  { ..., 1, 2, 0, 1, 2, ... } 
Rational  Q  Fractions with the numerator
and denominator integers, and
the denominator is not 0;
repeating decimal numbers. 
Irrational  H  Numbers that are not rational
numbers, like . 
Real  R  All numbers. 

© 1996 PrenticeHall, Inc.
 N W I Q R, and
H R
 Set Builder Notation
 Set builder notation is a way to express sets with out listing each element separately in roster form.

© 1996 PrenticeHall, Inc.

Set Builder Notation  Graphical Representation 
{ x  x > a }  
{ x  x a }  
{ x  a x < b }  
{ x  a x b }  
 Union and Intersection of Sets
 The union of two sets is a set containing all the elements from both sets.
 The intersection of two sets is a set containing the elements common to both sets.
 Symbols:
 means "union".
 means "intersection".
 Relation Symbols
 = means "is equal to": the lefthandside is equal to the righthandside.
 means "is not equal to": the lefthandside is not equal to the righthandside.
 < means "is less than": the lefthandside is less than the righthandside.
 means "is less than or equal to": the lefthandside is less than or equal to the righthandside.
 > means "is greater than": the lefthandside is greater than the righthandside.
 means "is greater than or equal to": the lefthandside is greater than or equal to the righthandside.
