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.
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