Consider the following recursive definition of a set X.
1. 12 ∈ X
2. 15 ∈ X
3. if x, y ∈ X, then x + y ∈ X
4. if x, y ∈ X, then x − y ∈ X
(a) Prove the following claim: Claim. Every member of X isdivisible by 3. Hint: You’ll want to do structural recursion on thedefinition of X, so there will be two base cases and two inductioncases.
(b) Prove the following claim: Claim. For every natural numbern, 3n ∈ X. Hint: In this case, you are not inducting on X, butinstead, you are inducting on the natural numbers.
Answer
If
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