Sets with the richest algebraic structure are known as fields. If the operations satisfy familiar arithmetic rules (such as associativity, commutativity, and distributivity) the set will have a particularly “rich” algebraic structure.
However, when a set has one or more operations (such as addition and multiplication) defined for its elements, it becomes very useful.
In itself a set is not very useful, being little more than a well-defined collection of mathematical objects. A definitive treatise, Modern Algebra, was written in 1930 by the Dutch mathematician Bartel van der Waerden, and the subject has had a deep effect on almost every branch of mathematics. As the techniques involved were similar, it seemed reasonable to consider the sets, rather than their elements, to be the objects of primary concern. The elements of the sets concerned could be numbers, functions, or some other objects. Modern algebra, also called abstract algebra, branch of mathematics concerned with the general algebraic structure of various sets (such as real numbers, complex numbers, matrices, and vector spaces), rather than rules and procedures for manipulating their individual elements.ĭuring the second half of the 19th century, various important mathematical advances led to the study of sets in which any two elements can be added or multiplied together to give a third element of the same set.
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