I'm having a hard time understanding how to show this relation is reflexive:
Let $X = \{1, 2, 3, 4, 5\} , Y = \{3, 4\}$. Define a relation $R$ on the power set of $X$ by
$A R B$ if $A \cup Y = B \cup Y$
How do I go about showing this is reflexive? The book says its reflexive because $A\cup Y = B\cup Y$
but wouldn't I want to see if the set containing the element of the relation $A R B$ is reflexive? How does this show anything? Furthermore, why do we not check $B \cup Y = B \cup Y$?
$A$ and $B$ are dummy set variables here. The definition says that for any two elements of $X$, which we call A and B and could be the same we have $A\ R\ B$ if and only if $A \cup Y=B \cup Y$. When you check reflexivity, $A$ and $B$ are the same because you want to prove $A\ R \ A$. To make sure you understand this relation, you should list the sets that are related to $\{1\}$, those related to $\{3\}$, and those related to $\{1,3\}$