Base amount

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Venn diagram of a set A in a basic set Ω

In mathematics, a basic set (also known as the universe ) refers to a set of all objects considered in a certain context. All quantities considered in this context are then subsets of this basic quantity. In individual cases, however, not all subsets of the basic set are also considered, for example in the case of a σ-algebra . In logic and in linguistics, the concept of the basic set corresponds to the universe of discourse ; in the predicate logic of the definition set .

The use of basic sets serves to avoid antinomies such as Russell's set antinomy . By choosing them appropriately, it is guaranteed that set operations such as averages and unions are defined and can only lead to meaningful (consistent) sets in the context of this.

Which objects can be included in the solution set for a given equation is crucially dependent on the basic set to which the equation relates.

In the case of an equation such as, for example , it is a proposition that is neither true nor false in itself. Only when concrete numbers are used instead of x does the statement form become a statement that is either true or false. When solving an equation, the number that is of interest to us is that makes the equation a true statement. The person who came up with this equation now also makes a further rule for the solver of this equation: One should only search within the natural numbers for an object or a number which makes a true statement from the equation. In other words: The basic set for the equation is prescribed in this case . As a result of this restriction, no number will be found that satisfies the equation. And therefore the solution set of the equation is empty.

If one agrees on another basic set, namely one that contains the number , e.g. B. the set of whole numbers or an even more comprehensive set of numbers, then the above equation has a solution, namely . The following applies to the solution set .

The choice of a basic set therefore has a considerable influence on whether an equation can be solved, and also on the number of elements in a possibly existing solution set. The same applies to inequalities and generally to forms of statements in which variables can occur.

See also