Let's start with three definitions:
- The Cartesian product [;X\times Y;] of two sets [;X\ Y;] is defined as the set of all ordered pairs such that their first and second terms belong respectively to [;X\ Y;]:
[;:\quad\quad X\times Y\ \ :=\ \ \{(x\ y)\ :\ x\in X\ \ \&\ \ y\in Y\};]
A set [;T;] is said to be a relational graph [; \Leftarrow : \Rightarrow ;] every element of [;T;] is an ordered pair.
An ordered triple [;\mathbf{T} := (X\ Y\ T);] is said to be a relation [; \Leftarrow : \Rightarrow\ \ T ;] is a relational graph such that [;T \subseteq X\times Y ;]. Such a relation [;\mathbf{T};] is said to be a relation in the sets [;X\ \ Y;], where [;X;] is called the domain of relation [;\mathbf{T};], and [;Y;] is called the codomain or the target of [;\mathbf{T};].
REMARK We could define a relation in fewer words: an ordered triple [;\mathbf{T} := (X\ Y\ T);] is said to be a relation [; \Leftarrow : \Rightarrow\ \ T \subseteq X\times Y ;].
We see that a relation is an ordered triple of the form:
(domain codomain rel_graph)
where rel_graph is a relational graph contained in the Cartesian product of the domain and codomain.
Observe that [;(X \ Y\ X\times\ Y);] is a relation in the sets [;X\ \ Y;]; its relational graph contains the relational graph of any other relation in the sets [;X\ \ Y;]. Also the empty set is a relational graph. It is contained in any other relational graph. Thus any relational graph of any relation in the sets [;X\ \ Y;]; is contained between these two relational graphs.
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