Comparison function

A comparison function is one of the six primitive dyadic scalar functions whose result on scalars is a Boolean determined by the numerical ordering of the two arguments. Comparisons are subject to comparison tolerance. The comparison functions can be subdivided into two equalities  , which depend only on whether the arguments are equal and not on the ordering of the arguments, and may be applied to characters, and four inequalities        , which depend on ordering and may not be applied to characters.

The six comparison functions are Less Than, Less Than or Equal , Equal to , Greater Than or Equal , Greater Than , and Not Equal to. You can take a closer look at how these six comparison functions behave in this APL Jupyter notebook.

Because comparison functions have Boolean results on Boolean arguments (they always have Boolean results), they are Boolean functions. In fact, the majority of nontrivial dyadic Boolean functions are comparison functions.

Classification
The result of a comparison function on real numbers is determined by their ordering, which falls into one of three possible cases: the left argument is smaller (more negative), the arguments are tolerantly equal, or the right argument is larger. A particular comparison function can thus be thought of as a choice of one Boolean value (0 or 1) for each of these three cases, which implies that there are eight total possibilities. Two of the possibilities—all 0 and all 1—are trivial and cannot be said to depend on the ordering of the arguments; they are not considered comparison functions. Every remaining possibility is one of the APL comparisons:

The traditional APL ordering of the comparison functions used above, which was used at least as early as APL\360, resembles a Gray code ordering of the values which define the comparison functions. Adjacent values differ by only a change of one bit, but the list is not a true Gray code because it excludes the all-0 and all-1 combinations.