Definition 2

The set $\mathscr{E}_z (z = x \text{ or } z = y)$, whose only elements are $x$ and $y$, is denoted by $\{x, y\}$.

The relation $z \in \{x, y\}$ is therefore equivalent to "$z = x$ or $z = y$"; it follows from C50 that $\{y, x\} = \{x, y\}$.

Let $R\{z\}$ be a relation and let $x, y$ be letters distinct from $z$. From the criteria C32, C33 (Chapter I, §4, no. 3), and C43 (Chapter I, §5, no. 1) it follows easily that the relation $(\exists z)((z \in \{x, y\})$ and $R\{z\})$ is equivalent to "$R\{x\}$ or $R\{y\}$"; consequently the relation $(\forall z)((z \in \{x, y\}) \Rightarrow R\{z\})$ is equivalent to "$R\{x\}$ and $R\{y\}$".

The set $\{x, x\}$, which is denoted simply by $\{x\}$, is called the set whose only element is $x$; the relation $z \in \{x\}$ is equivalent to $z = x$, and the relation $x \in \mathrm{X}$ is equivalent to $\{x\} \subset \mathrm{X}$.