- #1
thy
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Consider the set T defined recursively as follows:
• 2∈T,
• if x∈T and x>1,then x/2 ∈T,
• if x∈T and x>1,then x^2 ∈T,
• T contains no other element.
Use Structural Induction to write a detailed, carefully structured proof that ∀ x ∈ T, ∃ n ∈ N, x = 2n.
I'm not sure how to do this proof.
I assume ∀ x ∈ T, ∃ n ∈ N, x = 2n is true and try to prove then x/2 ∈T and x^2 ∈T?
• 2∈T,
• if x∈T and x>1,then x/2 ∈T,
• if x∈T and x>1,then x^2 ∈T,
• T contains no other element.
Use Structural Induction to write a detailed, carefully structured proof that ∀ x ∈ T, ∃ n ∈ N, x = 2n.
I'm not sure how to do this proof.
I assume ∀ x ∈ T, ∃ n ∈ N, x = 2n is true and try to prove then x/2 ∈T and x^2 ∈T?