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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!.
- Show that B(n) 2n-1 .

Bell numbers

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 ?(x, y) denotes the Hamming distance of x

and y,

Partial Orders (POSets)

Remark PO is a relation that is reflexive,

antisymmetric, and transitive

Cardinality

- A and B have the same cardinality (written

AB) iff there exists a bijection (bijective

function) from A to B. - if SN, we say S is countable. Else, S is

uncountable.

Cantors 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 is countably infinite if there

is a bijection f N ?A, - 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, R2, 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

- Prove that (a, 8) and (-8, a) each have the same

cardinality as (0, 8). - Prove that these sets have the same cardinality

(0, 1), (0, 1, 0, 1, (0, 1) U Z, R - Prove that given an infinite set A and a finite

set B, then A U B A.

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