Let A = {n ∈ Z | n = 5r for some integer r} and B = {m ∈ Z | m = 20s for some integer s}. a. Is A ⊆ B? Explain. b. Is B ⊆ A? Explain

Let $A = \{n\in\mathbb Z\ |\ n = 5r\ \text{for some integer}\ r\}$ and $B = \{m\in\mathbb Z\ |\ m = 20s\ \text{for some integer}\ s\}$.

a. Since $n=5=5\cdot1$ and $1\in\mathbb Z,$ we conclude that $n=5\in A$. On the other hand, $n=5\ne20\cdot s$ for each integer $s$, and therefore, $n=5\notin B$. We conclude that it is not true that $A\subseteq B.$

b. If $m\in B$ then $m = 20s$ for some integer $s$. It follows that $m=5(4s), 4s\in\mathbb Z$, and consequently, $m\in A$. Therefore, $B\subseteq A.$