An extension of the algebra of sets

Abstract

We shall explain the aim which leads us in the construction of an extended system of the algebra of sets1. The symbol 1. {*:?(*)} denoting the set of these and only these elements of domain of the variable x which satisfy the propositional condition (propositional function or form) ?9 (x)" is in com? mon use nowadays, so that it is adopted in school courses of mathematics in many countries, and in Poland as well. This condition will be said to define the set 1. However, if we admit propositional conditions which are meaningless for some values of their variables then we encounter some difficulties connected with the ex? pression 1. The formulae 2. {x : 9 (*)} = {x : 9 (*)}' 3. {x : 9 (s) v <|/ 0)} = {x : 9 (*)} u {x : ty (*)} do not hold for such conditions2. E.g. if we assume x to run over the set of all real numbers and the set to be the universe of discourse then we obtain