The Empty, Universal, and Identity Relations on a Set
Recall from the Relations on Sets page that if $X$ is a set then a relation $R$ on $X$ is a subset of the Cartesian product $X \times X$ where if $(x, y) \in R$ then we write $x \: R \: y$ and say "$x$ relates $y$ and if $(x, y) \not \in R$ then we write $x \: \not R \: y$ and say $x$ does not relate $y$.
We will now look at three rather basic relations on a set $X$.
| Definition: Let $X$ be a set. The Empty Relation $\emptyset$ on $X$ is defined to be the relation where for all $x, y \in X$ we have that $x \: \not R \: y$. |
For example, consider the set of integers $\mathbb{Z}$ and let $R$ be the relation such that for $x, y \in \mathbb{Z}$ we have that $x \: R \: y$ if both $x + y$ is even and $x + y$ is odd. Clearly the sum $x + y$ cannot both be even and odd, and so $R$ is the empty relation since for all $x, y \in X$ we have that $x \: \not R \: y$, i.e., $R = \emptyset$.
| Definition: Let $X$ be a set. The Universal Relation or Full Relation $\mathcal U$ on $\mathbb{Z}$ is defined to be the relation where for all $x, y \in X$ we have that $x \: \mathcal U \: y$. |
For example, consider the set of integers $\mathbb{Z}$ again. Define $R$ to be the relation such that for $x, y \in \mathbb{Z}$ we have that $x \: R \: y$ if $x + y \in \mathbb{Z}$. The sum of any two integers is always going to be an integer, and so for all $x, y \in \mathbb{Z}$ we have that $x \: R \: y$ so $R$ is the universal relation on $X$ so $R = \mathcal U$
| Definition: Let $X$ be a set. The Identity Relation $\mathcal I$ on $\mathbb{Z}$ is defined to be the relation where for all $x, y \in X$ we have that $x \: \mathcal I \: y$ if and only if $x = y$. |
For example, consider the set of integers $\mathbb{Z} \setminus \{ 0 \}$. Define $R$ to be the relation such that for $x, y \in \mathbb{Z}$ we have that $x \: R \: y$ if $\frac{x}{y} = 1$.
If $x \: R \: y$ then $\frac{x}{y} = 1$ so $x = y$. Conversely, if $x, y \in \mathbb{Z} \setminus \{0 \}$ and $x = y$ then $\frac{x}{y} = 1$ so $x \: R \: y$. Therefore $R$ is the identity relation on $\mathbb{Z} \setminus \{ 0 \}$ so $R = \mathcal I$.