# Completeness of Binary Relations

Completeness of binary relations often is defined as: The binary relation R of a set A is complete iff for any pair x,y ∈ A: xRy or yRx.

My question is: what does one mean by „pair“? To me it seems like one does not mean „pair“ as defined in math (ordered or unordered pair) but one means that colloquially.

E.g., if A={1,2,3}, the „pairs“ (as it is meant in the definition of completeness) are: 1 and 1 ; 1 and 2 ; 1 and 3 ; 2 and 2 ; 2 and 3 ; 3 and 3.

Am I right?

• "Pair" just means the two values, $x$ and $y$, taken together. Nothing more. And yes, in the full definition you can have $x = y$. Commented May 10, 2022 at 18:42
• Thanks for replying. To be sure what you mean: what I‘ve written above is correct right? Commented May 10, 2022 at 19:06

Thanks for replying. To be sure what you mean: what I‘ve written above is correct right?

Yes. The ordered pairs from the set $$A$$ are the elements of the Cartesian product $$A\times A$$.

So those from $$\{1,2,3\}$$ are:$$\begin{Bmatrix}(1,1),&(1,2),&(1,3),\\(2,1),&(2,2),&(2,3),\\(3,1),&(3,2),&(3,3)\end{Bmatrix}$$

Completeness requires that at least one of a pair in $$A\times A$$ or its ordered-inverse is in the relation. So your example is a complete relation:

$$\begin{Bmatrix}(1,1),&(1,2),&(1,3),\\&(2,2),&(2,3),\\&&(3,3)\end{Bmatrix}$$

As is this example:

$$\begin{Bmatrix}(1,1),&(1,2),&\\&(2,2),&(2,3),\\(3,1),&(3,2),&(3,3)\end{Bmatrix}$$