__Sets and Other Basic Concepts__Chapter 1

**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.