# Knowledge objects Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence. [← Previous](/en/knowledge/?page=6#results)[1](/en/knowledge/?page=1#results) … [6](/en/knowledge/?page=6#results) 7 [8](/en/knowledge/?page=8#results) … [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=8#results) 302 results · Page 7 of 13 Knowledge object ## [Event independence](/en/mko/mko-event-independence/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) Two events are independent when their intersection probability equals the product of their marginal probabilities; this definition does not rely on conditioning on a zero-probability event. mko-event-independence · Probability and statistics Knowledge object ## [Graph walks and simple paths](/en/mko/mko-graph-walk-simple-path/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) In a finite undirected simple graph, a walk requires each consecutive pair of vertices to be adjacent; a simple path additionally requires all vertices to be distinct. mko-graph-walk-simple-path · Combinatorics and discrete mathematics Knowledge object ## [Finite commutative ring](/en/mko/mko-finite-commutative-ring/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) A finite unital ring is commutative when multiplication commutes for every pair of elements; zero divisors may still exist. mko-finite-commutative-ring · Algebra Knowledge object ## [Finite two-sided ideal](/en/mko/mko-finite-two-sided-ideal/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) A subset of a finite ring is a two-sided ideal when it is an additive subgroup and absorbs multiplication by parent-ring elements on both sides. mko-finite-two-sided-ideal · Algebra Knowledge object ## [Real Banach space](/en/mko/mko-real-banach-space/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) A real normed vector space is a Banach space when it is complete in the metric induced by its own norm. mko-real-banach-space · Analysis Knowledge object ## [Real inner-product space](/en/mko/mko-real-inner-product-space/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) A symmetric, positive-definite real bilinear form is a real inner product; positive semidefiniteness or positive sampled values do not suffice. mko-real-inner-product-space · Analysis Knowledge object ## [Bounded linear operator](/en/mko/mko-bounded-linear-operator/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) A linear map between normed spaces is bounded when one nonnegative constant uniformly controls all output norms relative to input norms. mko-bounded-linear-operator · Analysis Knowledge object ## [Independence of random variables](/en/mko/mko-random-variable-independence/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) Under one finite joint law, independence requires every value-pair mass to factor as the product of its marginal masses. mko-random-variable-independence · Probability and statistics Knowledge object ## [Connected component](/en/mko/mko-connected-component/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) In a finite undirected simple graph, connected components are exactly the vertex equivalence classes defined by the existence of a path. mko-connected-component · Combinatorics and discrete mathematics Knowledge object ## [Finite integral domain](/en/mko/mko-finite-integral-domain/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) A finite commutative unital ring with 1 distinct from 0 is an integral domain when it has no nonzero zero divisors. mko-finite-integral-domain · Algebra Knowledge object ## [Polynomial ring over a finite commutative ring](/en/mko/mko-polynomial-ring-over-finite-commutative-ring/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) Ordinary polynomials over a finite commutative coefficient ring have finite-support coefficient sequences, but the collection over all finite degrees is still infinite. mko-polynomial-ring-over-finite-commutative-ring · Algebra Knowledge object ## [Inner-product-induced norm](/en/mko/mko-inner-product-induced-norm/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) On the same real inner-product space, the norm is the nonnegative square root of the self-inner-product; the triangle inequality uses the universal Cauchy–Schwarz certificate. mko-inner-product-induced-norm · Analysis Knowledge object ## [Finite conditional expectation](/en/mko/mko-conditional-expectation/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) Relative to a partition-generated sigma-algebra on a finite probability space, conditional expectation takes the weighted mean on each positive-mass block and preserves block integrals. mko-conditional-expectation · Probability and statistics Knowledge object ## [Finite filtration](/en/mko/mko-finite-filtration/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) A finite-time filtration is a sequence of sigma-algebras on the same probability space whose information increases with time. mko-finite-filtration · Probability and statistics Knowledge object ## [Independent identically distributed sequence](/en/mko/mko-iid-sequence/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) An IID sequence requires every variable to have the same distribution and every finite subfamily to be mutually independent. mko-iid-sequence · Probability and statistics Knowledge object ## [Equivalence class](/en/mko/mko-equivalence-class/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) For a fixed complete equivalence relation, the class of an element a consists of every carrier element related to a. mko-equivalence-class · Foundations and logic Knowledge object ## [Finite field](/en/mko/mko-finite-field/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) A finite commutative unital ring is a finite field when every nonzero element has a multiplicative inverse. mko-finite-field · Algebra Knowledge object ## [Real Hilbert space](/en/mko/mko-real-hilbert-space/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) A real inner-product space is Hilbert when it is complete in the norm and metric induced by that same inner product. mko-real-hilbert-space · Analysis Knowledge object ## [Finite martingale](/en/mko/mko-martingale/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) An integrable finite process adapted to the same filtration is a martingale when the next value's conditional expectation equals the current value almost surely at every step. mko-martingale · Probability and statistics Knowledge object ## [Quotient set](/en/mko/mko-quotient-set/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) The distinct equivalence classes of a complete equivalence relation form its quotient set and partition the original carrier. mko-quotient-set · Foundations and logic Knowledge object ## [Cardinality comparison of sets](/en/mko/mko-cardinality-comparison/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) An injection, surjection or bijection between explicit finite source and target sets supplies a corresponding finite cardinality-comparison certificate. mko-cardinality-comparison · Foundations and logic Knowledge object ## [Cumulative distribution function](/en/mko/mko-cumulative-distribution-function/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) For a real-valued random variable, F(t)=P(X≤t) is nondecreasing and right-continuous; in a finite discrete law, its jump is the point mass. mko-cumulative-distribution-function · Probability and statistics Knowledge object ## [Convergence in probability](/en/mko/mko-convergence-in-probability/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) X_n converges in probability to X when, for every epsilon>0, the probability of deviation greater than epsilon tends to zero. mko-convergence-in-probability · Probability and statistics Knowledge object ## [At-most-countable set](/en/mko/mko-at-most-countable-set/?from=%2Fen%2Fknowledge%2F%3Fpage%3D7) A set is at most countable if it is finite or has an explicit injection into the natural numbers; this is not the same as countably infinite. mko-at-most-countable-set · Foundations and logic [← Previous](/en/knowledge/?page=6#results)[1](/en/knowledge/?page=1#results) … [6](/en/knowledge/?page=6#results) 7 [8](/en/knowledge/?page=8#results) … [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=8#results) 302 results · Page 7 of 13