Verify my answer on induction definition for the set

Question

Write an induction definition for the set A={<a>, <b,a,b>,<a,b,a,b,a>, <b,a,b,a,b,a,b>,….}

basic: < a > ∈A

induction:if bxb∈A then a::bxb::a ∈A

if a::x then b::x::b ∈A

Answer 1

Your definition seems to work, roughly. Your use of $x$ and $::$ seems a bit inconsistent though. Here’s another way of writing it that is a bit more precise.

Basis:

$\langle a \rangle \in A$

Inductive step:

$\langle a::x \rangle \in A \implies \langle b::a::x::b \rangle \in A$

$\langle b::x \rangle \in A \implies \langle a::b::x::a \rangle \in A$