Question

Whichofthefollowingstatementsaretrueandwhicharefalse?Justifyyouranswerwitha shortprooforacounterexample. .2)Theoperation∗definedbyx∗y=log(xy)isabinaryoperationonS,whereSisthe set{x∈R|x>0}.

Answer 1

Answer on Question#54667 – Math – Linear Algebra

Question. Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.

2) The operation $*$ defined by $x * y = \log(xy)$ is a binary operation on $S$ where

$$S = \{x \in \mathbb{R} \mid x > 0\}.$$

Solution. Let $x = 1 > 0, y = \frac{1}{e} = e^{-1} > 0$. Then $\log(xy) = \log(e^{-1}) = -1 < 0 \Rightarrow$

$\Rightarrow \exists x, y \in S: \log(xy) \notin S \Rightarrow$ the operation $*$ is not defined on $S \Rightarrow$ the operation $*$ is not a binary operation on $S$.

Answer. False.

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