In the text and in the homework, the notation A+A' and A*A' is used. What does this mean?

First, $A$ and $A'$ are two subsets of the rational numbers. $A+A'$ represents any rational number which can be expressed as the sum of one in $A$ and another in $A'$. For example, if $A$ consists of all rationals less than 2, and $A'$ of all rationals less than 5, then $A+A'$ consists of all rationals less than 7.

Similarly, $A\cdot A'$ represents the set of all rationals which can be expressed as the product of something in $A$ with something in $A'$. This gets a bit tricky, since two negative numbers multiply to a positive number. Thus, if $A$ is the numbers less than 2 and $A'$ those less than 5, then $A\cdot A'$ consists of *all* rational numbers. Resolving this trickiness with negative numbers is the purpose of one of the homework problems.