- 108 Views
- Uploaded on

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

**An Image/Link below is provided (as is) to download presentation**

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 countable if it is either finite or countably infinite.

- 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 . . .

Arrive at a contradiction.

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

Answers:

- (3,3); 3 < x < 3
- (,2] [2,); x 2 or x 2
- [1,7]; 1 x 7
- 4. (,7) (3,); x < 7 or x > 3
- 5. (1,4); 1 < x < 4

Download Presentation

Connecting to Server..