1 City (1990 pop. 9,942), seat of Montgomery co., SE Kans., on the Verdigris River, near the Okla. line, in an important oil-producing area where corn and wheat are also grown. Light aircraft, motor vehicle parts, cement, and printing and publishing are important industries; natural gas is distributed. The town was founded (1869) on a former Osage reservation. It boomed with the discovery of natural gas in 1881 and oil in 1903.
2 City (1990 pop. 112,301), seat of Jackson co., W Mo., a suburb of Kansas City; inc. 1849. Its manufactures include machinery, building materials, apparel, foods, paper products, and ordnance. Soybeans, corn, and sorghum are grown, and there is dairying and natural-gas production in the area. In the 1830s and 40s, Independence was the starting point for expeditions over the Santa Fe Trail Santa Fe Trail, important caravan route of the W United States, extending c.780 mi (1,260 km) from Independence, Mo., SW to Santa Fe, N.Mex. Independence and Westport, Mo., were the chief points where wagons, teams, and supplies were obtained.
..... Click the link for more information. , the Oregon Trail Oregon Trail, overland emigrant route in the United States from the Missouri River to the Columbia River country (all of which was then called Oregon). The pioneers by wagon train did not, however, follow any single narrow route.
..... Click the link for more information. , and the California Trail. A group of Mormons settled there in 1831, and the city is the world headquarters of the Community of Christ Community of Christ, formerly the Reorganized Church of Jesus Christ of Latter Day Saints, religious group that regards itself as the successor of the church founded by Joseph Smith.
..... Click the link for more information. (formerly the Reorganized Church of Jesus Christ of Latter Day Saints). Independence was the home of President Harry S. Truman Truman, Harry S., 1884–1972, 33d President of the United States, b. Lamar, Mo.
Early Life and Political Career
He grew up on a farm near Independence, Mo., worked at various jobs, and tended the family farm.
..... Click the link for more information. and is the seat of the Harry S. Truman Library and Museum, on whose grounds the former president is buried. Other points of interest include the old county jail and museum (1859; restored); the old county courthouse (1825; restored); and nearby Fort Osage (1808; reconstructed). Park Univ. has a campus in Independence.
Independence
City (pop., 2000: 113,288), western Missouri, U.S. Settled in 1827, it served as the starting point for the Santa Fe Trail and the Oregon Trail and was a rendezvous for wagon trains during the California gold rush. Home of a Mormon colony (1831–33), it is now the world headquarters of the Community of Christ (formerly Reorganized Church of Jesus Christ of Latter Day Saints). It was occupied by Union troops during the American Civil War and was the scene of two skirmishes with Confederates. The hometown of Pres. Harry Truman, it is the site of the Harry S. Truman Library and Museum.
Independence
Independence
Independence
in logic, the property of a sentence of some theory or formula of a calculus such that neither the sentence itself nor its negation can be derived from a given system of sentences (for example, a system of axioms) or, correspondingly, from a conjunction of given formulas, respectively.
A sentence can be shown to be independent of a given system of axioms by proving the consistency of two systems of axioms that can be obtained from the addition, respectively, of the given proposition and its negation to the given system of axioms. Independence is also related to the property of deductive completeness of axiomatic theories. If a consistent system of axioms is deductively complete, a contradiction results when any proposition (independent of the system) of the given theory is added as an axiom to the system. In speaking of the independence of intuitively formulated sentences, “derivability” is understood intuitively, “in accordance with the laws of logic.” On the other hand, strictly defined rules of inference (the question of the independence of which can also be raised) are always fixed when considering formal calculi.
It is possible to speak of “expressive” independence in a way analogous to the “deductive” independence described above. In this case, a concept (term) is said to be independent of a given system of concepts (terms) if it cannot be defined solely by means of these concepts (terms), although again, as above, it is assumed that a set of rules of definition has been fixed with respect to which the question of independence can be raised. The term “independence” (in both senses) is, finally, also applied to sets of sentences (formulas) or concepts (terms). A set is said to be independent (and also nonredundant, or minimal) if every one of its members is independent of the remaining members in the sense defined above. A number of highly important results concerning independence have been obtained in the axiomatic theory of sets and in mathematical logic.
IU. A. GASTEV
Independence
one of the most important concepts in probability theory. We give as an example the definition of the independence of two random events. Let A and B be two random events, and P(A) and P(B) their probabilities. The conditional probability P(Bǀ A) of the event B under the condition that A occurs is defined by the equality
where P(A&B) is the probability that A and B occur simultaneously. The event B is said to be independent of A if
(*) P(B\A) = P(B)
Equation (*) can be written in a form symmetric in A and B:
P(A&B) = P(A)P(B)
from which it is evident that if B is independent of A, then A is independent of B. Thus, we may simply speak of the independence of two events.
The specific meaning of this definition of independence can be clarified in the following manner. It is known that the probability of an event is expressed by the frequency of its occurrence. Therefore, if a large number N of trials is carried out, then the frequency with which the event B appears in all ¿V trials and the frequency with which it appears in those trials in which the event A occurs will be approximately equal. Thus, the independence of events indicates either that there is no relation between the occurrence of these events or that the relation is not essential. Thus, the event in which a randomly selected person has a last name beginning, for example, with the letter “A” and the event that this person will win the next drawing of a lottery are independent.
Pairwise and mutual independence are distinguished in defining the independence of several (more than two) events. The events A1, A2, . . .,An are said to be pairwise independent if any two of them are independent in the sense of the definition given above and are mutually independent if the probability that any of them occurs is independent of the occurrence of an arbitrary combination of the other events.
The concept of independence is also extended to random variables. The random variables X and Y are said to be independent if for any two intervals Δ1 and Δ2, the events that the variable X belongs to Δ2 and that Y belongs to Δ2 are independent. Highly important schemes in probability theory are based on the hypothesis that various events or random variables are independent.
REFERENCES
Gnedenko, B. V. Kurs teorii veroiatnostei, 4th ed. Moscow, 1965.Feller, W. Vvedenie v teoriiu veroiatnostei i ee prilozheniia, 2nd ed. Moscow, 1964. (Translated from English.)
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