 An interval formula_2 is a proper subinterval of formula_3 if formula_2 is a proper subset of formula_3.The terms segment and interval have been employed in the literature in two essentially opposite ways, resulting in ambiguity when these terms are used.The "Encyclopedia of Mathematics" defines "interval" (without a qualifier) to exclude both endpoints (i.e., open interval) and "segment" to include both endpoints (i.e., closed interval), while Rudin's "Principles of Mathematical Analysis" calls sets of the form ["a", "b"] "intervals" and sets of the form ("a", "b") "segments" throughout.