CS 390 Solutions to Homework 7

1. Basis Step: Prove that $P(\Lambda )$ holds.
Inductive Step: Assuming that $P(w)$ holds for an arbitrary string w, prove that $P(aw)$ and $P(bw)$ hold.

2. The children of $q$ are the members of the set $\delta (q, \Lambda )$.

3. Basis Step: Prove that $S \subseteq \Lambda (S \cup T )$.
Note that S is the basis of \Lambda (S). So in the basis step we prove that the elements of S i.e. the seeds of \Lambda(S), belong to \Lambda(S U T).

Inductive Step: Assuming that for an arbitrary $q \in \Lambda (S)$, $q \in \Lambda (S \cup T )$ holds, prove that $\delta (q, \Lambda ) \subseteq \Lambda (S \cup T )$.
In the inductive step we need to prove that if any element of \Lambda(S) belongs to \Lambda(S U T), then its children also belong to \Lambda(S U T). Since the children of an arbitrary element q are the members of $\delta (q, \Lambda )$, what we need is to prove that $\delta (q, \Lambda ) \subseteq \Lambda (S \cup T )$.