The Class of Series-Parallel Graphs. Part III

In this paper A, B, a, b, c, d, e, f , g, h denote sets. One can prove the following three propositions: (1) idA↾B = idA ∩ [:B, B :]. (2) id{a,b,c,d} = {〈a, a〉, 〈b, b〉, 〈c, c〉, 〈d, d〉}. (3) [: {a, b, c, d}, {e, f, g, h} :] = {〈a, e〉, 〈a, f〉, 〈b, e〉, 〈b, f〉, 〈a, g〉, 〈a, h〉, 〈b, g〉, 〈b, h〉} ∪ {〈c, e〉, 〈c, f〉, 〈d, e〉, 〈d, f〉, 〈c, g〉, 〈c, h〉, 〈d, g〉, 〈d, h〉}. Let X, Y be trivial sets. Observe that every relation between X and Y is trivial. We now state the proposition (4) For every trivial set X and for every binary relation R on X such that R is non empty there exists a set x such that R = {〈x, x〉}.