Lower Tolerance . Preliminaries to Wroclaw Taxonomy 1

The paper introduces some preliminary notions concerning the Wroclaw taxonomy according to [14]. The classifications and tolerances are defined and considered w.r.t. sets and metric spaces. We prove theorems showing various classifications based on tolerances. In this paper A, X denote non empty sets, f denotes a partial function from [: X, X :] to R, and a denotes a real number. Let us observe that there exists a real number which is non negative. We now state a number of propositions: (1) For every finite sequence p and for every natural number k such that k + 1 ∈ dom p and k / ∈ dom p holds k = 0. (2) Let p be a finite sequence and i, j be natural numbers. Suppose i ∈ dom p and j ∈ dom p and for every natural number k such that k ∈ dom p and k + 1 ∈ dom p holds p(k) = p(k + 1). Then p(i) = p(j). (3) For every set X and for every binary relation R on X such that R is reflexive in X holds dom R = X.