Is it correct that — opposed to general relations and functions — equivalence relations and bijective functions can be defined without reference to ordered pairs? Especially, do the following definitions capture the usual notions of equivalence and bijection?
Definition: $X$ is an equivalence relation on set $Y$ if
$X \subset \mathcal{P}(Y)$
$(\forall x \in X)\ |x| = 1 \vee |x| = 2$
$(\forall y \in Y)\ \lbrace y \rbrace \in X$
$(\forall x,y,z \in Y)\ \lbrace x,y \rbrace \in X \wedge \lbrace y,z \rbrace \in X \rightarrow \lbrace x,z \rbrace \in X$
Definition: $X$ is a bijection between sets $Y$ and $Z$ if
$(\forall x \in X)(\exists y \in Y)(\exists z \in Z)\ \lbrace y,z\rbrace = x$
$(\forall y \in Y)(\forall z \in Z)(\exists x \in X)\ \lbrace y,z\rbrace = x$
Or is there a mistake in one of these definitions?