Set comparison

Stable Numerics Subroutine Library
Programming Reference Manual
Version 1.0 DD-00006-010

5.5.1 Set comparison

bool seq(const interval<T>&, const interval<T>&);
bool sne(const interval<T>&, const interval<T>&);
bool sle(const interval<T>&, const interval<T>&);
bool sge(const interval<T>&, const interval<T>&);
bool slt(const interval<T>&, const interval<T>&);
bool sgt(const interval<T>&, const interval<T>&);

Set comparison operators treat the given interval numbers as sets and provide exact comparisons, empty intervals of $∅$ are explicitly considered valid. That is for $X,Y ∈IR$ and $□∈ {=,<,≤,>,≥}$ :

X□Y ≡(∀a∈ X,∃b∈ Y :a □b)∧(∃a∈ X,∀b ∈Y :a ̸=b)

A special case is for inequality $̸=$ which is the same as above but using a disjunction instead of a conjunction, that is:

X ̸= Y ≡(∀a∈ X,∃b∈ Y :a ̸=b)∨ (∃a ∈X,∀b ∈Y :a̸= b)

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