How do I go about determining whether a relation is reflexive? I've been given these relations and I've been told to determine whether they're reflexive and I know the definition of reflexive but I don't really understand it.
$R=\{(x,y)\ \in\ \mathbb Z^2\ |\ x^2+y^2\ \text{is odd}\}$
$R=\{(x,y)\ \in\ \mathbb Q^2\ |\ xy \ge 0\}$
 A: If you have a relation 
$$R=\{(x,y)\   | \mbox{ something something}\}$$
Checking reflexivity simply means: If we set $x=y$ is "something something" always true (in the give set)?
Thus for the first relation, you have to check if $x^2+x^2$ is always odd for $x$ integer.
For the second relation, you have to check if $xx \geq 0$ is always true for $x$ rational.
A: A relation $R$ is said to be reflexive if $aRa$ for all $a$ in the domain.
So in
$$R=\{(x,y)\ \in \ \mathbb Z^2\ |\ x^2+y^2 is\:odd\}$$
$R$ will be reflexive if when $(a,a) \in \mathbb Z^2$ satisfies the relation for all $a \in \mathbb Z$. That is, if $2a^2$ is odd for all $a \in \mathbb Z$.
This is obviously false and thus the relation is not reflexive. Similarly, the other part can be solved.
A: For the second relation: $$ R = \{(x,y) \in \mathbb{Q} | xy \geq 0 \} $$
if the relationship is reflexive then x = y. Therefore you can rewrite xy as $x^{2}$ or $y^{2} $. Now you just have to answer whether a squared rational will always be greater than or equal to zero.
