DISCRETE MATHDefine a set S recursively as follows: I. Base:
$$
1 \in S , 3 \in S , 5 \in S , 7 \in S , 9 \in S
$$
, II. RECURSION: If
$$
s \in S
$$
and
$$
t \in S
$$
then, a.
$$
S t \in S
$$
, b.
$$
2 s \in S
$$
, c.
$$
4 s \in S
$$
, d.
$$
6 s \in S
$$
, e.
$$
8 s \in S
$$
. III. RESTRICTION: Nothing is in S other than objects defined in I and II above. Use structural induction to prove that every string in S represents an odd integer.