Relations

Question 1

partial order simple definition

Answer 1

The axioms for a non-strict partial order state that the relation ≤ is reflexive, antisymmetric, and transitive.

Question 2

partial order full definition

Answer 2

A (non-strict) partial order[2] is a homogeneous binary relation ≤ over a set P satisfying particular axioms which are discussed below. When a ≤ b, we say that a is related to b. (This does not imply that b is also related to a, because the relation need not be symmetric.)

The axioms for a non-strict partial order state that the relation ≤ is reflexive, antisymmetric, and transitive. That is, for all a, b, and c in P, it must satisfy:

a ≤ a (reflexivity: every element is related to itself).
if a ≤ b and b ≤ a, then a = b (antisymmetry: two distinct elements cannot be related in both directions).
if a ≤ b and b ≤ c, then a ≤ c (transitivity: if a first element is related to a second element, and, in turn, that element is related to a third element, then the first element is related to the third element).

Question 3

a preorder

Answer 3

a binary relation that is reflexive and transitive

Question 4

Reflexive

Answer 4

a relation R over set P is reflexive if for all a in P aRa, meaning a is related to itself