• Terminology

• Set: a collection of objects

• Element or Member of a set: an object belonging to the set

• Three ways to designate sets:

• word description

ex: the set of odd counting numbers between 2 and 12

• the listing method

ex: {3, 5, 7, 9, 11}

• set-builder notation

ex: {x | x ∈N, x is odd, and x < 12}

**• Notes**:

• use curly braces to designate sets,

• use commas to separate set elements

• the variable in the set -builder notation doesn’t have to be x.

• use ellipses (. . . ) to indicate a continuation of a pattern

established before the ellipses

ex: {1, 2, 3, 4, . . . , 100}

• N Natural or Counting numbers: {1, 2, 3, . . . }

• W Whole Numbers: {0, 1, 2, 3, . . . }

• I Integers: {. . . , -3, -2, -1, 0, 1, 2, 3, . . . }

• Q Rational numbers :

• - Real Numbers : { x | x is a number that can be written as a

decimal }

• Irrational numbers: { x | x is a real number and x cannot be

written as a quotient of integers }.

Examples are:

• Empty Set: { }

## Notes

• The symbols { x | x . . . } is read "x such that x . . .
has some

property

• The symbol ∈means is "an element of"

• Any rational number can be written as either a

TERMINATING decimal ( like 0.5, 0.333, or 0.8578966)

or a

REPEATING decimal ( like

• The decimal representation of an irrational number never

terminates and never repeats

• The set { } is not empty, but is a set which contains the empty

set

## Set Cardinality

• Cardinality of a set: the number of distinct elements in
the set

• textbook: n(A) - or we can use |A|

• If the cardinality of a set is a particular whole number, we call

that set a finite set

• If a set is too large to ever finish the counting process, it is called

an infinite set

• Well-Defined set: one for which we can determine
membership,

i.e., given any arbitrary value we can determine conclusively

whether or not that value is in the set

## Set Membership

• Well-Defined means that given a set and an object, we
can

determine if the set contains that object

## Set Equality

• Set Equality: the sets A and B are equal (written A = B)
provided:

• every element of A is an element of B, and

• every element of B is an element of A

i.e., if they contain exactly the same elements