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3 Binary Operations - Arkansas Tech University

Arkansas Tech UniversityMATH 4033: Elementary Modern AlgebraDr. Marcel B. Finan3 Binary OperationsWe are used to addition and multiplication of real numbers. These operationscombine two real numbers to generate a unique single real number. So wecan look at these Operations as functions on the setR R={(a, b) :a Rand b R}defined by+ :R R R(a, b) a+band :R R R(a, b) a bThese Operations are examples of a Binary operation. The general definitionof a Binary operation is as operationon a set S is a mapping that assigns to each orderedpair of elements ofSa uniquely determined element is, :S S Sis a mapping. The setSis said to beclosedunder the operation .The image (a, b) will be denoted bya and multiplication are Binary Operations on the setZof integersso that this set is closed under these Operations . However,Zis not closedunder the operation of division since 1 2 is not an ordered pair statement in Definition is critical.

For example, consider the set M(Z) of all functions from the set of integers into itself. Then the operation of composition is a binary operation on M(Z). Consider the two functions f(n) = 2n and g(n) = ˆ n 2 if n is even 4 if n is odd 5

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