1. Relations, Functions, and Countability
Set Theory
Relations, Functions, and Countability

2. Relations Show that B(n) ≤ . Show that B(n) ≤ n!.
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

3. 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,

4. Partial Orders (POSets)
Remark
PO is a relation that is reflexive, antisymmetric, and transitive

5. 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.

6. 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

7. 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.

8. 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

9. Diagonal Argument

10. 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

11. 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.

12. One-to-One Correspondence
Prove that (a, ∞) and (−∞, a) each have the same cardinality as (0, ∞).
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|.