Set Theory Relations Functions and Countability Relations Let

Set Theory Relations, Functions, and Countability

Relations • Let B(n) denote the number of equivalence relations on n elements. • Show that B(n) ≤ n!. • Show that B(n) ≥ 2 n− 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 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, 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|.