talmængder

Talmängder, the Swedish term for sets of numbers, are mathematical collections whose elements are numbers. In mathematics, a talmängd is any collection of numbers drawn from a fixed universe, such as the natural numbers or the real numbers. The concept is central to number theory and to algebra, allowing precise formulation of properties that various numbers may share.

Common number systems are the natural numbers N, integers Z, rational numbers Q, real numbers R and complex numbers C. These form a chain of inclusions, with N as a subset of Z, which is a subset of Q, which is a subset of R, which is a subset of C, though conventions vary on whether 0 or 1 belongs to N. A talmängd is usually specified by listing its elements or by a rule, for example A = {x ∈ N | x is even} or A = {2n | n ∈ Z}.

Operations on talmängder include union, intersection and set difference, and they can be combined with logical conditions. Important notions are finiteness versus infinitude, and countability. A talmängd is finite if it contains a finite number of elements; it is infinite otherwise. Among infinite sets, N, Z and Q are countable, while R and C are uncountable.

Common examples include the set of even numbers E = {2n | n ∈ Z}, the set of odd numbers O = {2n+1 | n ∈ Z}, and the set of prime numbers P = {p ∈ Z | p > 1 and only divisible by ±1 and ±p}. More generally, any subset of a given number system is itself a talmängd and may carry algebraic structure such as closure under addition or multiplication.

Talmängder provide a formal language for describing numerical properties, solving equations, and proving theorems in mathematics.