Properties of the Intervals of Real Numbers

The articles [5], [6], [1], [2], and [3] provide the notation and terminology for this paper. In this paperx, y, a, b, a1, b1, a2, b2 are extended real numbers. The following four propositions are true: (1) If x 6=−∞ andx 6= +∞ andx≤ y, then 0R ≤ y−x. (2) If x =−∞ andy =−∞ andx = +∞ andy = +∞ andx≤ y, then 0R ≤ y−x. (8)1 For all extended real numbers a, b, c such thatb 6= −∞ andb 6= +∞ anda = −∞ and c =−∞ anda = +∞ andc = +∞ holds(c−b)+(b−a) = c−a. (9) inf{a1,a2} ≤ a1 and inf{a1,a2} ≤ a2 anda1 ≤ sup{a1,a2} anda2 ≤ sup{a1,a2}. Let a, b be extended real numbers. The functor [a,b] yields a subset of R and is defined as follows: (Def. 1) For every extended real number x holdsx∈ [a,b] iff a≤ x andx≤ b andx∈ R. The functor]a,b[ yields a subset of R and is defined as follows: (Def. 2) For every extended real number x holdsx∈ ]a,b[ iff a < x andx < b andx∈ R. The functor]a,b] yielding a subset of R is defined by: (Def. 3) For every extended real number x holdsx∈ ]a,b] iff a < x andx≤ b andx∈ R. The functor[a,b[ yielding a subset of R is defined by: (Def. 4) For every extended real number x holdsx∈ [a,b[ iff a≤ x andx < b andx∈ R. Let I1 be a subset of R. We say that I1 is open interval if and only if: (Def. 5) There exist extended real numbers a, b such thata≤ b andI1 = ]a,b[. We say thatI1 is closed interval if and only if: 1 The propositions (3)–(7) have been removed.