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Inclusion–exclusion principle. In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as. where A and B are two finite sets and | S | indicates the cardinality of a ...
Exchange interaction is the main physical effect responsible for ferromagnetism, and has no classical analogue. For bosons, the exchange symmetry makes them bunch together, and the exchange interaction takes the form of an effective attraction that causes identical particles to be found closer together, as in Bose–Einstein condensation.
In proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used. The rule of sum, rule of product, and inclusion–exclusion principle are often used for enumerative purposes. Bijective proofs are utilized to demonstrate that two sets have the same number of elements.
We need to consider only the outer 3p 2 electrons, for which it can be shown (see term symbols) that the possible terms allowed by the Pauli exclusion principle are 1 D , 3 P , and 1 S. Hund's first rule now states that the ground state term is 3 P, which has S = 1. The superscript 3 is the value of the multiplicity = 2S + 1 = 3.
In host–guest chemistry, an inclusion compound (also known as an inclusion complex) is a chemical complex in which one chemical compound (the "host") has a cavity into which a "guest" compound can be accommodated. The interaction between the host and guest involves purely van der Waals bonding. The definition of inclusion compounds is very ...
Probability theory. The standard probability axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. [1] These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. [2]
Inclusion compounds are often molecules, whereas clathrates are typically polymeric. Intercalation compounds are not 3-dimensional, unlike clathrate compounds. [7] According to IUPAC, clathrates are "Inclusion compounds in which the guest molecule is in a cage formed by the host molecule or by a lattice of host molecules." [8]
There are two main categories of identical particles: bosons, which can share quantum states, and fermions, which cannot (as described by the Pauli exclusion principle). Examples of bosons are photons, gluons, phonons, helium-4 nuclei and all mesons. Examples of fermions are electrons, neutrinos, quarks, protons, neutrons, and helium-3 nuclei.