Independence (mathematics)

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In mathematics, the term independence carries a variety of distinct meanings:

In probability theory, to say that two events are statistically independent intuitively means that knowing whether or not one of them occurs makes it neither more probable nor less probable that the other occurs. For example, the event of getting a "1" when a die is thrown and the event of getting a "1" the second time it is thrown are independent. Formally, n events A₁, ..., A_n are independent if Pr(A₁ and ... and A_n) = Pr(A₁) × ... × Pr(A_n). An infinite collection of events is independent if every finite subcollection is independent. Real-valued random variables X₁, ..., X_n independent, provided that for all real numbers s₁, ..., s_n, the events [X₁ ≤ s₁], ..., [X_n ≤ s_n] are independent events. Independence of infinitely many random variables is defined analogously.

A set of vectors is said to be linearly independent provided that the only representation of zero as a linear combination of these vectors is the trivial combination where each vector is multiplied by zero. If every element of a vector space S can be written a linear combination of the set of given vectors, those vectors are said to span S. A collection of linearly independent vectors that span S is said to be a basis for S. For example, the vectors (1,0,0), (0,1,0), and (0,0,1) are linearly independent, and they form a basis for three-dimensional Euclidean space.

The argument to a function is the independent variable. This is an entirely different concept from those of statistical independence or linear algebraic independence.

A proposition P is said to logically independent of a set of axioms A, provided that there exists a consistent mathematical model where the proposition P and the axioms of A are true, and also there exists a consistent model where "not P" and the axioms of A are true. Equivalently, this means that if one assumes the axioms in A to be true (but does not assume more), one would not get a contradiction by also assuming that P is true, and, also, would not get a contradiction by instead assuming P to be false.

Paul Cohen proved that the "continuum hypothesis" concerning the cardinality of the set of real numbers is a proposition that is independent of the usual model for arithmetic. This settled problem number 1 on David Hilbert's famous list of mathematical problems.

References

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