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標題:

ALGEBRA, BINARY OPERATION

發問:

Let S be an arbitrary non-empty set, x,y,z is in S, and suppose that * is an associative binary operation on S. Prove: if x*y=y*x and x*z=z*x, then x*(y*x)=(y*z)*x. Please provide full proof step and clear explanation Thankssssssss

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最佳解答:

x*(y*x)=(y*z)*x is not true when z =/= x and S = the set of positive integers and * = usual multiplication. I think your question is to prove x*(y*z) = (y*z)*x. x*(y*z) = (x*y)*z (since the binary operation * is associative) = (y*x)*z (since x*y = y*x from assumption) = y*(x*z) (since the binary operation * is associative) = y*(z*x) (since x*z=z*x from assumption) = (y*z)*x (since the binary operation * is associative)

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