Relations

Goal

We want to study binary relations on a set & the properties and representations that are implied.

Definition

A relation on the set S can literally be defined as anything, there is no rigid mathematical rule for it. A relation simply maps some subset of elements of one set to some subset of elements in the other.

A relation on a set S is a subset of the cartesian product of sets.

To study the nature of the relations between two sets, we like to talk about binary relations:

Cartesian product of sets $A$ and $B$:

$A \times B = \{ (a_1, b_1), (a_2, b_2), ..., (a_n, b_n)\}$

such that $a_i$ and $b_i$ are elements in $A$ and $B$ respectively.

So a binary relation $R$ on the set $S$ is holistically $R \subseteq S_1 \times S_2$, and more precisely, the binary relation R maps elements in one set to those in another to essentially give us a set of 2-tuples.

We can also extrapolate and have N-ary relations that map multiple sets together like $R \subseteq A \times B \times C$.

Furthermore we like to study binary relations on a set. Meaning that we attempt to study the relations that the elements of a set have on itself.

When we do this, some properties and representations will be visible. We will attempt to study these properties and representations.