36 Logic as a Tool If a ∈ [X ]G then a−1 ∈ [X ]G If a, b ∈ [X ]G then a ◦ b ∈ [X ]G Exercise: re-state the definition above as an inductive definition 1.4.2 Induction principles and proofs by induction With every inductive definition, a scheme for proofs by induction can be associated The construction of this scheme is uniform from the inductive definition, as illustrated in the following 1.4.2.1 Induction on the words in an alphabet We begin with a principle of induction that allows us to prove properties of all words in a given alphabet Given an alphabet A, let P be a property of words in A such that: The empty string has the property P If the word w in A has the property P and a ∈ A, then the word wa has the property P Then, every word w in A has the property P 1.4.2.2 Induction on natural numbers We can now formulate the well-known principle of mathematical induction on natural numbers in terms of the formal definition of natural numbers given above Let P be a property of natural numbers such that: has the property P For every natural number n, if n has the property P then Sn has the property P Then every natural number n has the property P Here is the same principle, stated in set-theoretic terms: Let P be a set of natural numbers such that: ∈ P For every natural number n, if n ∈ P then Sn ∈ P Then every natural number n is in P, that is, P = N 1.4.2.3 Structural induction on propositional formulae Following the same pattern, we can now state a principle of induction that allows us to prove properties of propositional formulae Note that this principle is obtained almost