Q: Consider the following relation on $A \times A$ where $A = \Bbb Z$ $ \times $ {$1, 2, 3,...$}:
$R = $ {$\langle\langle p, q\rangle, \langle p', q'\rangle\rangle$ $|$ $p \centerdot q' = q \centerdot p'$}
Is it an equivalence relation? (The claim should be properly justified!)
What confuses me are the marks on elements $p'$ and $q'$. Does it have anything to do with complements or are they elements of the second set?
The pair $\langle p', q' \rangle$ is an element of $A$, which means that $p'$ is an element of $\mathbb Z$ and $q'$ is an element of $\{1, 2, 3, \ldots \}$. It is just a way to produce new variable names, without using new letters.
The relation could equally well have been written down as $$ R = \{\langle\langle p, q\rangle, \langle r, s\rangle \rangle \mid p \times s = q \times r\}. $$ Note that $R \subseteq A \times A$. Normally we would call $R$ a (binary) relation on $A$, not a relation on $A \times A$.