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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.

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cardinality of a set
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
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 | = .
slide3
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)

slide4
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

slide5
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
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.

slide7
A set S is countable if it is either finite or countably infinite.
  • A set S is uncountable if it is not countable.
examples
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|.

slide9
3. I (the irrational numbers) and 

 (the real numbers) are uncountable sets;

that is

|I| > |N| and | | > |N|.

some facts
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|.
slide11
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, 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”

slide14
is not a rational number, i.e., is an irrational number.

Proof:

Suppose is a rational number. Then

. . .

other examples of irrational numbers
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

slide16
Algebraic numbers –

roots of polynomials with integer coefficients.

Transcendental numbers –

irrational numbers that are not algebraic.

slide17
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
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
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
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
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