The union of intervals in constructive mathematics

Abstract

Constructive Real Numbers (CRNs) are the concept of real numbers used in Constructive Mathematics, where the objects are thought of as being generated by a computer program. This paper focuses on the property of unions in the constructive end, which means that we study the union of intervals from the viewpoint of Constructive Mathematics. In the end, we prove that for t < u ≤ v < w (t, u, v, w are CRN), then it is always true that [t, v] ∪ [u, w] ⊂ [t, w] but generally it is not true that [t, w] ⊂ [t, v] ∪ [u, w]. However, for t < u < v < w(t, u, v, w are CRN), it is always true that [t, w] = [t, v] ∪ [u, w].