Abstract Algebra: An Introduction by Thomas W Hungerford

By Thomas W Hungerford

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6) where a,, a2, ... are the terms of the sequence. According to our definition, the terms may be any, not necessarily distinct, objects. 6) and a further sequence bI.... , n). 6) n is called the number of the terms of the sequence; (8;6) itself is also called a sequence of n terms. Further ai is called the i`h term of the sequence, i itself the index of this term. and call it an (ordered) n-tuple. , In the special cases n = 2, 3, 4 the terms pair (pair of elements), triple, and quadruple are also used.

A semiordered set , in which a

If T is a subset of S, then aT denotes the set of distinct elements aoc (a E T). Ta and T° are to be interpreted similarly. If the operation is denoted by ax or a°, then the operator is sometimes called a coefficient or an exponent. Compositions and operations have a similarity of content and form, so that one s accustomed to calling them inner and outer compositions respectively. g. a o a, but the additive type of notation is rarely used for operators. Usually a set S is assigned an operator domain O only if some compositions have already been defined in S.

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