CmSc 365 Theory of Computation
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چکیده
Finite and Infinite Sets. Countability. Proof Techniques (Chapter 1, Sections 1.4, 1.5) Finite sets finite number of elements can be counted Infinite sets infinite number of elements Equinumerous sets: Sets A and B are equinumerous if there is a bijection f : A → B E.G.: A = {1,2,3}, B = {a,b,c} Countably infinite sets: A set is countably infinite if it is equinumerous with N (the set of natural numbers) Countable sets Finite sets and countably infinite sets are countable. Sets that are not countable, are called uncountable. Proving countability/uncountability of sets: Three basic techniques: a. finding a bijection to the set of natural numbers b. applying the dovetailing technique c. using the diagonalization principle These techniques may be used in a combined way, together with direct proof and proof by contradiction A. Bijection Example 1: The set of all squares of natural numbers is countably infinite A = {k : k ∈ N}, i.e. A = {1, 2, 4, 9, ...} We can construct a bijection f : A → B f(k) = k Hence A is equinumerous to N.
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