A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a , b \in W$, the numbers $a - b$ and $a b$ are also in $W$. Let $S = \left\{ \left. \frac { m } { 2 ^ { n } } \right\rvert\, m , n \text{ integers} \right\}$ and $T = \left\{ \left. \frac { p } { q } \right\rvert\, p , q \text{ integers}, q \text{ odd} \right\}$. Then: (a) neither $S$ nor $T$ is a ring (b) $S$ is a ring, $T$ is not a ring. (c) $T$ is a ring, $S$ is not a ring. (d) both $S$ and $T$ are rings.
(d) Verify using the given definition of a ring.
A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a , b \in W$, the numbers $a - b$ and $a b$ are also in $W$. Let $S = \left\{ \left. \frac { m } { 2 ^ { n } } \right\rvert\, m , n \text{ integers} \right\}$ and $T = \left\{ \left. \frac { p } { q } \right\rvert\, p , q \text{ integers}, q \text{ odd} \right\}$. Then:\\
(a) neither $S$ nor $T$ is a ring\\
(b) $S$ is a ring, $T$ is not a ring.\\
(c) $T$ is a ring, $S$ is not a ring.\\
(d) both $S$ and $T$ are rings.