LetthesetSbedefinedrecursivelyasfollows:Base: 1∈SRecursion: ifs∈S,thena)0s∈Sandb)1s∈SRestriction:NothingisinSotherthantheobjectsdefinedabove.(a) Prove by structural induction that ALL elements in S endwith a 1. Note that to carry out a proof by structural inductionrequires two steps:(i) Show that each element/object in the BASE of the recursivedefinition satisfies the property. (i i ) show that for each rulein the RECURSION, if the rule is applied to elements/ob- jectsalready in the set that satisfy the property, then the objectsdefined by the rule also satisfy the property.
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