Induction & Recursion · step 5 of 13

Inductively defined sets & languages

An inductive definition specifies a set by base elements, closure rules, and the crucial minimality clause: nothing else belongs. That last clause is what licenses structural induction — without it the definition would describe any superset. Formal languages are usually presented this way, and balanced brackets is the canonical example.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Give the base elementsThe starting points, such as the empty string ε or the atomic formulas.
Give the closure rulesHow to build new elements from old ones. Each rule becomes a case in later induction proofs.
State minimalityAdd "and nothing else is in S". This is not decoration — it is what makes the set unique.
Prove membership by derivationShow a construction sequence from base elements to the target using only the rules.

Worked example

Let S be defined by: ε ∈ S; if x ∈ S then (x) ∈ S; if x, y ∈ S then xy ∈ S. Show ()(()) ∈ S.

ε ∈ S by the base rule.
Apply rule 2 to ε: () ∈ S.
Apply rule 2 to (): (()) ∈ S.
Apply rule 3 to () and (()): ()(()) ∈ S.

Answer. ()(()) ∈ S via the derivation ε → () → (()) then concatenation — S is exactly the set of balanced bracket strings.

Where marks get dropped

These are the specific errors that cost credit on inductively defined sets & languages questions — QED's rubric penalises each of them separately.

Omitting the minimality clause, which leaves the definition satisfied by many different sets.
Applying a rule to something not yet shown to be in the set. Each derivation step must use previously established members.
Confusing the language with the set of all strings over the alphabet. The rules generate a proper subset.

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Inductively defined sets & languages — frequently asked questions

Why is the minimality clause necessary?

Because the base plus rules only say the set is CLOSED under the rules. The set of all strings is closed too — minimality picks out the intended smallest one.

How do I prove something is NOT in the set?

By structural induction: find a property held by every base element and preserved by every rule, then show the candidate lacks it. For brackets, "equal counts and no prefix with more right than left" works.

What is the connection to grammars?

A context-free grammar is exactly an inductive definition of a language, with productions as the closure rules and derivations as membership proofs.

The rest of Induction & Recursion

Mathematical, strong & structural induction; recursion. Each subtopic below has its own method, worked example and mark-losing traps.

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