My book says that both of these statements are true. I don't understand why the first is true. Suppose $A = \{1, 2, 3, 4\}$, and define the relations as follows:
$R_1 = \{(1, 1), (2, 2)\} $
$R_2 = \{(3, 3), (4, 4)\}$
These two relations are reflexive, but the intersection is the empty set. The empty set is not reflexive for a non-empty set $A$. Why am I wrong here?
On
Neither of these relations is reflexive. For a relation $R$ on a set $A$ to be reflexive, we must have $(x,x)\in R$ for all $x\in A$. Your relations satisfy this only for some of the elements of the set $A=\{1,2,3,4\}$.
It is true that your $R_1$ is reflexive as a relation on the set $B=\{1,2\}$ and $R_2$ is reflexive as a relation on the set $C=\{3,4\}$. But the statement in your book requires $R_1$ and $R_2$ to be reflexive relations on the same set $A$.
On
Correct me if I am wrong, but a relation is reflexive if $(x,x) \in R$ for all $x\in A$. Hence neither of the two relations that you give are reflexive.
To see that the intersection of two reflexive relations is reflexive, suppose that $R_1$ and $R_2$ are reflexive. Then $$R_1 \cap R_2 = \{ (x,y) : (x,y) \in R_1 \land (x,y) \in R_2 \}.$$ But we know that for any $x\in A$, we have $(x,x)\in R_1$, and $(x,x) \in R_2$, therefore $(x,x) \in R_1 \cap R_2$. Therefore $R_1 \cap R_2$ is reflexive.
The union is similar.
Neither of these relations is reflexive: Reflexivity means that for all elements $a \in A, (a,a) \in R$. In this case, $(3,3) \notin R_1$ and $(1,1) \notin R_2$.