Cardinality

Set Theory

Symbols & Terminology

A set is a collection of objects. The objects belonging to the set are called the elements, or members of the set.

Designating Sets

Sets are designated using:

1. word description

2. the listing method 3. set-builder notation

Designating Sets

Word description

The set of even counting numbers less than 10

Listing method

{2, 4, 6, 8}

Set-builder notation

{x|x is an even counting number less than 10}

Designating Sets

Sets are commonly given names (capital letters). A = {1, 2, 3, 4}

The set containing no elements is called the empty set (null set) and denoted by { } or Ø.

To show 2 is an element of set A use the symbol 2{1,2,3,4}

a{1,2,3,4}

Sets of Numbers

Natural (counting) {1, 2, 3, 4, ...} Whole numbers {0, 1, 2, 3, 4, ...} Integers {...,–3, –2, –1, 0, 1, 2, 3, ...}

Rational numbers

May be written as a terminating decimal, like 0.25, or a repeating decimal like 0.333...

Irrational {x | x is not expressible as a quotient of integers} Decimal representations never terminate and never repeat.

Real numbers {x | x can be expressed as a decimal}

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Cardinality

The number of elements in a set is called the cardinal number, or cardinality of the set.

The symbol n(A), read “n of A,” represents the cardinal number of set A.

Finite & Infinite Sets

If the cardinal number of a set is a particular

whole number, we call that set a finite set.

Whenever a set is so large that its cardinal number is not found among the whole numbers, we call that set an infinite set.

Equality of Sets

Set A is equal to set B provided the following two conditions are met:

1. Every element of A is an element of B, and

2. Every element of B is an element of A.

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