jelölfüggvény

A jelölfüggvény, or indicator function, in mathematics is a function that tells you whether an element belongs to a particular subset of a set. It is usually defined on a set X and a subset A of X. The indicator function, often denoted as $1_A$ or $I_A$, takes a value of 1 if the element is in the subset A, and 0 otherwise.

Formally, for a set X and a subset A $\subseteq$ X, the indicator function $1_A: X \to \{0, 1\}$ is defined as:

$1_A(x) = 1$ if $x \in A$

$1_A(x) = 0$ if $x \notin A$

Indicator functions are widely used in probability theory, statistics, and set theory. They provide a convenient way to express set operations and properties using algebraic means. For example, the intersection of two sets A and B can be represented by the product of their indicator functions: $1_{A \cap B}(x) = 1_A(x) \cdot 1_B(x)$. Similarly, the union of two sets can be represented by $1_{A \cup B}(x) = 1_A(x) + 1_B(x) - 1_A(x) \cdot 1_B(x)$, or more simply, $1_{A \cup B}(x) = \max(1_A(x), 1_B(x))$. The complement of a set A within X can be expressed as $1_{A^c}(x) = 1 - 1_A(x)$.

In probability, the expected value of an indicator function is the probability of the event represented by the subset. That is, for a random variable X and an event A, $E[1_A(X)] = P(X \in A)$. This property makes indicator functions very useful for calculating probabilities and deriving statistical properties. They are a fundamental tool for simplifying complex set-theoretic statements and integrating them into analytical frameworks.