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 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
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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
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•
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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|.