Set Theory

Relations, Functions, and Countability

Relations

• Let

B

(

n

) denote the number of equivalence relations on n elements. • Show that B(n) ≤ .

• Show that B(n) ≤ n!.

Bell numbers

• Show that B(n) ≥ 2 n−1 .

Functions and Equivalence Relations

Remark

Equivalence relation is a relation that is

reflexive

,

symmetric

, and

transitive

• Suppose that: • Is a function?

• Which of the following is an equivalence relation?

where Δ(x, y) denotes the

Hamming distance

of x and y,

Cardinality

•

A

and

B have the same cardinality

(written |

A

|=|

B

|) iff there exists a bijection (bijective function) from

A

to

B

.

• if |S|=|N|, we say S is

countable

. Else, S is

uncountable

.

Cantor’s Theorem

• The power set of any set

A

has a strictly greater cardinality than that of

A

.

• There is no bijection from a set to its power set. • Proof By contradiction

Countability

• An infinite set

A f:

ℕ

→A

, is countably infinite if there is a bijection • A set is

countable

if it finite or countably infinite.

Countable Sets

• Any subset of a countable set • The set of integers, algebraic/rational numbers • The union of two/finnite sum of countable sets • Cartesian product of a finite number of countable sets • The set of all finite subsets of N; • Set of binary strings

Diagonal Argument

Uncountable Sets

•

R

,

R

2 , P(

N

) • The intervals [0,1), [0, 1], (0, 1) • The set of all real numbers; • The set of all functions from N to {0, 1}; • The set of functions N → N; • Any set having an uncountable subset

Transfinite

Cardinal Numbers

• Cardinality of a

finite

set is simply the number of elements in the set. • Cardinalities of

infinite

sets are not natural numbers, but are special objects called

transfinite cardinal numbers

•  0 :  |

N

|, is the

first transfinite cardinal

number.

•

continuum hypothesis

claims that |

R

|=  1 , the

second transfinite cardinal.

One-to-One Correspondence

1.

Prove that (a, ∞) and (−∞, a) each have the same cardinality as (0, ∞).

2.

Prove that these sets have the same cardinality: (0, 1), (0, 1], [0, 1], (0, 1) U Z, R 3.

Prove that given an infinite set A and a finite set B, then |A U B| = |A|.