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

• Suppose that:• Is a function?

• Which of the following is an equivalence relation?

RemarkEquivalence relation is a relation that is reflexive, symmetric, and transitive

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 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

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|.