2.Properties and Special Relations

3.Functions as relations

An *n-ary* relation is a subset of the cartesian product of $n$ sets. Most common are binary relations. Given a set $A$ and a set $B$ we can form a binary relation $R$ as a subset of $A \times B$. For example if we have
$$
A = \{1,2,3\}
$$
and
$$
B = \{a,b\}
$$
we can form the following relations
$$
R_1 = \{ (1,a),(2,a) \}
$$
or
$$
R_2 = \{ (1,a),(2,b),(3,a) \}
$$
However the relation
$$
R_{not} = \{ (1,1),(a,1) \}
$$
is not a subset of $A \times B$ since $(1,1) \not \in A \times B$ and $(a,1) \not \in A \times B$.

For a relation, $R$, we tend to describe that two elements, $a,b$, are related by $a\ R\ b$ - meaning $a$ is related to $b$ by relation $R$. The order is important. $a\ R\ b$ doesn't necessarily imply that $b\ R\ a$.

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