Question

(c) Show that $S/I$consists of exactly two distinct co-sets.

Short answer

It is proved $S/I$consists of exactly two distinct co-sets.

Step-by-step solution

Consider the two sets

Consider the two sets in$S/I$are$I$ and$I+1$.

Show that S/I consists of exactly two distinct co-sets

Assume that $\frac{a}{b}\in S-I$, such that $a=2k+1$for $k\in \mathbb{Z}$.

As $2|-b+1$, then

$\frac{a}{b}=\frac{2k+1}{b}=\frac{2k-b+1}{b}+1\in I+1$

Thus, two sets in $S/I$are $I$and $I+1$.

Hence, it is proved$S/I$ consists of exactly two distinct co-sets.