Computability and Logic (5th Edition, 2007) – Solutions to
Problems (Part A & B) – Boolos
Enumerable set - answers-One whose elements can be arranged in a single list
Enumerable set (formal) - answers-A is enumerable ↔there is f s.t. f enumerates A ↔f is a
bijection and f; N --> A ↔A is countable
Cardinality (of A and B) - answers-1. ||A|| = ||B|| iff there is a bijection f:A --> B 2. ||A|| ≤
||B|| iff there is an injection f:A-->B
Finite set - answers-A set is finite if it has the same cardinality as set {0,...,n} for n∈N
Countable set - answers-A set is countable iff ||A|| ≤N0
Cantor's theorem - answers-For any set A, ||A|| < ||p(A)||
Diagonalization - answers-Mighty important method to prove a set is uncountable
Doubler Turing Machine - answers-Uses 11 states to double the input
TM-computable function - answers-f; N-->N is T-computable iff there exists M st M computes f
<> f is the k-ary function computed by M if when M starts in a standard position:
1. If f(n1,...,nk) = m, then M eventually halts in a standard position, and
2. If (fn1,...,nk) ↑, then M never halts or halts in a non-standard position
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