xy yields True when x_i≤y_i is True for all i=1,…,n:

xy yields False when x_i>y_i is False for any i=1,…,n:

Represent a vector inequality:

When v is replaced by numerical vector space elements, the inequality gives True or False:

The cone is also given by :

The cone is also given by :

The cuboid is also given by :

Determine if all of the elements in a vector are non-negative:

Determine if all components are less than or equal to 1:

!xy does not imply xy:

For each component, !x_i≤y_i does imply x_i>y_i:

Compare the components of two matrices:

Compare symmetric matrices:

Represent the condition that Norm[{x,y}]<=1:

Represent the condition that :

Show where for non-negative x,y with α between 0 and 1:

VectorLessEqual is compatible with a vector space operation:

Adding vectors to both sides for any vector :

Multiplying by positive constants for any :

xy is a (non-strict) partial order, i.e. reflexive, antisymmetric and transitive:

Reflexive, i.e. for all elements :

Antisymmetric, i.e. if and then :

Transitive, i.e. if and then :

x_κy are partial orders but not total orders, so there are incomparable elements:

Neither nor is true, because and are incomparable elements:

The set of vectors and . These are the comparable elements to :

Vector orders are partial orders, so the negation of is not equivalent to :

Here both and are false:

Visualize and . The difference of these sets consists of incomparable elements: