Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.

1. Introduction to set theory and to methodology and philosophy of
mathematics and computer programming
Inductive definitions and structural induction
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of February 14, 2017

2. successor(n) = n + 1
Natural numbers can be constructed from 0 by repeated applications of successor.
To prove that every natural number has a property P, one can use mathematical
induction.
Sets of other mathematical objects can be defined by repeated applications of some
constructs to certain base objects – these are inductive definitions .
To prove that every object in such a set has a property P, one can use
structural induction .

3. Definition
Let Prop be a set of propositional symbols.
The set of propositional formulas over Prop ,
denoted Form(Prop) , is defined as follows
in an inductive definition.
1. , ⊥ ∈ Form(Prop).
2. If p ∈ Prop, then p ∈ Form(Prop).
3. If A ∈ Form(Prop), then ¬A ∈ Form(Prop).
4. If A, B ∈ Form(Prop), then (A ∧ B), (A ∨ B), (A → B) ∈ Form(Prop).
5. Nothing belongs to Form(Prop) unless it can be constructed using the preceding
rules.

4. Definition
Let Prop be a set of propositional symbols.
The set of propositional formulas over Prop ,
denoted Form(Prop) , is defined as follows
in an inductive definition.
1. , ⊥ ∈ Form(Prop). Base clause
2. If p ∈ Prop, then p ∈ Form(Prop). Base clause
3. If A ∈ Form(Prop), then ¬A ∈ Form(Prop). Inductive clause
4. If A, B ∈ Form(Prop), then (A ∧ B), (A ∨ B), (A → B) ∈ Form(Prop). Inductive
clause
5. Nothing belongs to Form(Prop) unless it can be constructed using the preceding
rules. Restriction clause

5. Definition
Let Prop be a set of propositional symbols.
The set of propositional formulas over Prop ,
denoted Form(Prop) , is defined as follows
1. , ⊥ ∈ Form(Prop).
2. If p ∈ Prop, then p ∈ Form(Prop).
3. If A ∈ Form(Prop), then ¬A ∈ Form(Prop).
4. If A, B ∈ Form(Prop), then (A ∧ B), (A ∨ B), (A → B) ∈ Form(Prop).
5. Nothing belongs to Form(Prop) unless it can be constructed using the preceding
rules.
Notice the parentheses in (A ∧ B), (A ∨ B), (A → B).

6. Proposition
Let Prop be a set of propositional symbols.
Every formula X from Form(Prop) contains equal number of left and right parentheses.
Proof by structural induction
Base case: X is .
has equal number of left and right parentheses, namely 0.
Base case: X is ⊥.
⊥ has equal number of left and right parentheses, namely 0.
Base case: X is a propositional symbol p from Prop.
p has equal number of left and right parentheses, namely 0.
Inductive case: X is ¬A, for some A ∈ Form(Prop).
By inductive hypothesis, A has equal number of left and right
parentheses. X, which is ¬A, has the same numbers of left and right
parentheses as A. So X has equal number of left and right parentheses.

7. Inductive case: X is (A ∧ B), for some A, B ∈ Form(Prop).
By inductive hypothesis, A has equal number of left and right
parentheses, say m, and B has equal number of left and right
parentheses, say n. Then, X, which is (A ∧ B), has the m + n + 1 left
parentheses, and the same number of right parentheses.
Inductive case: X is (A ∨ B), for some A, B ∈ Form(Prop).
By inductive hypothesis, A has equal number of left and right
parentheses, say m, and B has equal number of left and right
parentheses, say n. Then, X, which is (A ∨ B), has the m + n + 1 left
parentheses, and the same number of right parentheses.
Inductive case: X is (A → B), for some A, B ∈ Form(Prop).
By inductive hypothesis, A has equal number of left and right
parentheses, say m, and B has equal number of left and right
parentheses, say n. Then, X, which is (A → B), has the m + n + 1 left
parentheses, and the same number of right parentheses.

8. Exercise
Prove by structural induction that for every A ∈ Form(Prop),
if A does not involve , ⊥
then the number of left parentheses in A is
one less than the number of occurrences of propositional symbols in A.

9. Definition
Let A be a propositional formula.
The set of subformulas of A , denoted subformulas(A) , is defined as follows
in an inductive definition.
1. A ∈ subformulas(A).
2. If (B ∧ C) ∈ subformulas(A), then B, C ∈ subformulas(A).
3. If (B ∨ C) ∈ subformulas(A), then B, C ∈ subformulas(A).
4. If (B → C) ∈ subformulas(A), then B, C ∈ subformulas(A).
5. If ¬B ∈ subformulas(A), then B ∈ subformulas(A).
6. Nothing belongs to subformulas(A) unless it can be constructed using the
preceding rules.

10. Definition
Let A be a propositional formula.
The set of subformulas of A , denoted subformulas(A) , is defined as follows
in an inductive definition.
1. A ∈ subformulas(A). Base clause
2. If (B ∧ C) ∈ subformulas(A), then B, C ∈ subformulas(A). Inductive clause
3. If (B ∨ C) ∈ subformulas(A), then B, C ∈ subformulas(A). Inductive clause
4. If (B → C) ∈ subformulas(A), then B, C ∈ subformulas(A). Inductive clause
5. If ¬B ∈ subformulas(A), then B ∈ subformulas(A). Inductive clause
6. Nothing belongs to subformulas(A) unless it can be constructed using the
preceding rules. Restriction clause

11. Definition
Let A be a propositional formula.
The set of subformulas of A , denoted subformulas(A) , is defined as follows
in an inductive definition.
1. A ∈ subformulas(A).
2. If (B ∧ C) ∈ subformulas(A), then B, C ∈ subformulas(A).
3. If (B ∨ C) ∈ subformulas(A), then B, C ∈ subformulas(A).
4. If (B → C) ∈ subformulas(A), then B, C ∈ subformulas(A).
5. If ¬B ∈ subformulas(A), then B ∈ subformulas(A).
6. Nothing belongs to subformulas(A) unless it can be constructed using the
preceding rules.
Exercise: Let A be the formula (p → (q ∨ ¬p)). Find the set subformulas(A).

12. Definition
Let A be a propositional formula.
The set of subformulas of A , denoted subformulas(A) , is defined as follows
in an inductive definition.
1. A ∈ subformulas(A).
2. If (B ∧ C) ∈ subformulas(A), then B, C ∈ subformulas(A).
3. If (B ∨ C) ∈ subformulas(A), then B, C ∈ subformulas(A).
4. If (B → C) ∈ subformulas(A), then B, C ∈ subformulas(A).
5. If ¬B ∈ subformulas(A), then B ∈ subformulas(A).
6. Nothing belongs to subformulas(A) unless it can be constructed using the
preceding rules.
Exercise: Let A be the formula (p → (q ∨ ¬p)). Find the set subformulas(A).
Answer: {(p → (q ∨ ¬p)), p, (q ∨ ¬p), q, ¬p}