Math 101 Intermediate Algebra
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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 Prentice-Hall, 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 Prentice-Hall, 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 left-hand-side is equal to the right-hand-side.

  means "is not equal to": the left-hand-side is not equal to the right-hand-side.

<  means "is less than": the left-hand-side is less than the right-hand-side.

  means "is less than or equal to": the left-hand-side is less than or equal to the right-hand-side.

>  means "is greater than": the left-hand-side is greater than the right-hand-side.

  means "is greater than or equal to": the left-hand-side is greater than or equal to the right-hand-side.