Answer to Question #167365 in Linear Algebra for Nikhil Singh

Question #167365

Check whether or not * is a binary operation on S={x∈R| x>0}, where x*y=| In(xy)|


1
Expert's answer
2021-03-01T07:10:20-0500

By definition of a binary operation, we should have for any x,ySx,y\in S, xySx*y\in S. But if we take xS,y=1xx\in S, y=\frac{1}{x} (that always exists, as x0x\neq 0), we obtain xy=ln(x1x)=0Sx*y=|\ln(x\cdot \frac{1}{x})|=|0|\notin S. Thus the image of * is not contained in SS, so * is not a binary operation.


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