EDB — 23X

[23X]A relation \(R\) between elements of \(A\) is said to be:

• reflexive if \(xRx\) for any \(x∈ A\);

• irreflexive or anti-reflexive if \(\lnot xRx\) for any \(x∈ A\);

• symmetric if \(xRy\) implies \(yRx\) for any \(x,y∈ A\);

• antisymmetric if \(aRb\) and \(bRa\) imply \(a=b\), for any \(a,b∈ A\);

• trichotomous if for all \(x,y \in A\) one and exactly one of \(xRy\), \(yRx\) and \(x = y\) holds;

• transitive if \(xRy\) and \(yRz\) imply \(xRz\), for any \(x,y,z∈ A\).

A relation \(R\) between elements of \(A\) and elements of \(B\) is said to be:

• injective (also called left-unique) if \(xRy\) and \(zRy\) imply \(x = z\), for any \(x,z∈ A,y∈ B\);

• functional (also called right-unique) if \(xRy\) and \(xRz\) imply \(y = z\), for any \(x∈ A,y,z∈ B\); such a binary relation is called a “partial function” (see also [1YR]↺↻,[01P]↺↻);

• total (also called “left-total”) if for any \(x∈ A\) there is a \(y∈ B\) such that \(xRy\);

• surjective (also called “right-total”) if for any \(y∈ B\) there is a \(x∈ A\) such that \(xRy\).