Cardinality of a Set

1 / 22

# Cardinality of a Set - PowerPoint PPT Presentation

Cardinality of a Set. “The number of elements in a set.” Let A be a set. If A =  (the empty set), then the cardinality of A is 0. b. If A has exactly n elements, n a natural number, then the cardinality of A is n. The set A is a finite set.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Cardinality of a Set' - oistin

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Cardinality of a Set
• “The number of elements in a set.”
• Let A be a set.
• If A =  (the empty set), then the cardinality of A is 0.
• b. If A has exactly n elements, n a natural number, then the cardinality of A is n. The set A is a finite set.
• c. Otherwise, A is an infinite set.
Notation
• The cardinality of a set A is denoted by | A |.
• If A =  , then | A |= 0.
• If A has exactly n elements, then | A | = n.
• c. If A is an infinite set, then | A | = .
Examples:

A = {2, 3, 5, 7, 11, 13, 17, 19}; | A | = 8

A = N (natural numbers); | N | = 

A = Q (rational numbers); | Q | = 

A = {2n | n is an integer}; | A | = 

(the set of even integers)

DEFINITION: Let A and B be sets. Then,

|A| = |B| if and only if there is a one-to-one

correspondence between the elements of A and the

elements of B.

Examples:

1. A = {1, 2, 3, 4, 5}

B = {a, e, i, o, u}

1 a, 2 e, 3 i, 4 o, 5 u; |B| = 5

2. A = N (the natural numbers)

B = {2n | n is a natural number} (the even natural

numbers)

n 2n is a one-to one correspondence between

A and B. Therefore, |A| = |B|; |B| = .

3. A = N (the natural numbers)

C = {2n 1 | n is a natural number} (the odd

natural numbers)

n 2n 1 is a one-to one correspondence between

A and C. Therefore, |A| = |C|; |C| = .

Countable Sets

DEFINITIONS:

1. A set S is finite if there is a one-to-one correspondence between it and the set

{1, 2, 3, . . ., n} for some natural number n.

2. A set S is countably infinite if there is a one-to-one correspondence between it and the natural numbers N.

• A set S is uncountable if it is not countable.
Examples:

1. A = {1, 2, 3, 4, 5, 6, 7},

 = {a, b, c, d, . . . x, y, z}

are finite sets; |A| = 7, | | = 26 .

2. N (the natural numbers), Z (the integers), and Q (the rational numbers) are countably infnite sets;

that is, |Q| = |Z| = |N|.

3. I (the irrational numbers) and 

 (the real numbers) are uncountable sets;

that is

|I| > |N| and | | > |N|.

Some Facts:
• A set S is finite if and only if for any proper subset A  S, |A| < |S|; that is, “proper subsets of a finite set have fewer elements.”
• Suppose that A and B are infinite sets and A  B. If B is countably infinite then A is countably infinite and |A| = |B|.
3. Every subset of a countable set is countable.
• If A and B are countable sets, then A  B
• is a countable set.
Irrational Numbers, Real Numbers

Irrational numbers: “points on the real line

that are not rational points”; decimals that

are neither repeating nor terminating.

Real numbers: “rationals”  “irrationals”

is not a rational number, i.e., is an irrational number.

Proof:

Suppose is a rational number. Then

. . .

Other examples ofirrational numbers:

Square roots of rational numbers that are not

perfect squares.

Cube roots of rational numbers that are not

perfect cubes.

And so on.

  3.14159, e 2.7182182845

Algebraic numbers –

roots of polynomials with integer coefficients.

Transcendental numbers –

irrational numbers that are not algebraic.

THEOREM: The real numbers are

uncountable!

Proof: Consider the real numbers on the

interval [0,1]. Suppose they are countable.

Then . . .

COROLLARY: The irrational numbers

are uncountable.

Proof: Real numbers: “rationals”  “irrationals”

Absolute Value

DEFINITION: Let a be a real number. The absolute value of a, denoted |a|, is given by

Geometric interpretation: |a| is the distance on the real number line from the point a to the origin 0.

Absolute value inequalities
• Find the real numbers x that satisfy:
• 1. |x| < 3
• 2. |x| 2
• 3. |x  3|  4
• 4. |x + 2| > 5
• 5. |2x  3| < 5