Recursion and induction

1
Recursion and induction

• definition of a (universal) set by induction
  (recursion)
• definition of an operation (given a universe) by
  induction (recursion)
• proof of a property by induction
• Recursive definitions
• Step 1 Basic building blocks (atoms)
• Step 2 Construct complex from simpler
• Step 3 There is nothing else (only for sets,
  not for operations)
• Inductive proof
• Step 1 Prove for atoms
• Step 2 Reduce (proof of) complex to
  simpler (other direction as inductive
  definitions)

2
Recursive definitions

• Example 1 Suppose we take U N.
• We can define the set E of even naturals by
• 0 ? E (the atom is 0)
• If n ? E, then n2 ? E.
• Nothing else is in E.
• E is the smallest subset of U containing the
  atoms and closed under the operations. Without
  smallest, e.g. E 0,2,4,5,6,7,8, would
  also satisfy the first two clauses.
• Example 2 Suppose U 1, 2, 3, 4, 5.
• Let f U ? U be given by f(n) 6-n.
• What subset do we get from the recursive
  definition below?
• Our only atom is 3
• If n belongs to the subset, so does f(n)
• It is the smallest subset containing 3 and
  closed under f.

3
More recursive definitions

• Example 3 Suppose we take U N.
• Let s N ? N be given by s(n) n1 (the
  successor function). What subset is recursively
  defined if we specify
• the atom is 113
• whenever k belongs to it, so does s(k)?
• Example 4 Suppose U N?N.
• What subset of N?N do we get from the recursive
  definition below?
• atom is 0, 0.
• If m, n belongs, so does s(m), s(s(n))
• Example 5 Take U N?N. The points on the line
  y 3x
• The atom is 0, 0.
• If x, y belongs, so does s(x), s(s(s(y))) .

4
Defining functions recursively

• We have recursively defined subsets of N?N. We
  now define functions (operations) on N.
• Example The factorial function is defined by
• 0! 1
• (n1)! (n1) ?n!
• In the usual function notation, we have defined a
  function Fact N ? N by stipulating that
• Fact(0) 1 Fact(s(n)) s(n)?Fact(n).
• Or in yet other words, we define the subset of
  N?N called Fact by specifying that
• our atom is 0, 1
• if n, k ? Fact then s(n), s(n)?k ? Fact.
• (And nothing else is in Fact)

5
Induction

• For every recursively defined set there is a
  corresponding induction principle.
• Lets focus on N.
• N can be described as the smallest subset of R
  containing atom 0 and closed under s.
• So if a subset X of N has the properties
• 0 ? X
• if n ? X, then s(n) n1 ? X
• then that subset must be the whole of N.
• Principle of induction for N
• Let X ? N. If
• 0 ? X and
• for every n ? X, s(n) ? X,
• then X N.
• Similarly for other atoms and recursive
  operations!

6
Examples of inductive proofs

• Prove that, for every n ?? N,
• 0 1 2 n n(n1)/2.
• Proof
• Basis 0 0(01)/2
• Induction 0 1 2 n (n1) (inductive
  hypothesis) n(n1)/2 (n1) (basic
  algebra) (n1)(n2)/2 (end of proof)
• Prove that, for every finite set X and n ? N,
• if X n, then ?(X) 2n
• Proof (see book)

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Variations on induction

• 1. Other recursively defined sets.
• A 4, 5, 6, may be defined by saying that 4
  ? A and that if n ? A, then s(n) ? A.
• Prove that, for every n 4,
• n! gt 2n.
• 2. Other formulations of the principle.
• Suppose X ? N and X has the property that
• if 0?X, 1?X, 2?X, , n?X then s(n)?X.
• Then X N.
• This is generally called strong induction (for
  the natural numbers, similarly for other
  inductively defined sets)
• Prove that every integer k gt 1 is a product of
  prime numbers.