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For subsets $X$ and $Y$ of $\mathbb{R}$, define: $X*Y ≡ \{ z\in ℝ \mid\text{there exist }x\in X, y\in Y\text{ such that }z = xy \}.$

Then define for $0\lt a\lt b$, $[a,b] = \{x\in\mathbb{R}\mid a\leq x\leq b\}$ and let $X=[a,b]$.

and they ask me to find $Y\subseteq \mathbb{R}$ such that $X*Y=X$.

any ideas?

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    A simpler way to write the definition is $X*Y=\{xy \mid x\in X\text{ and }y\in Y\}$.2012-05-03

1 Answers 1

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Hint If $y$ is any element of $Y$, then $ay \in X*Y$ thus

$a \leq ay \leq b$

and $by \in X*y$ thus

$a \leq by \leq b \,.$

Since $a,b$ are positive, the inequalities $a \leq ay$ and $by \leq b$ tell you that there is only one possible value $y$ can take....

P.S. The problem is more interesting, and more complicated if $a,b$ have opposite signs. There are then multiple solutions for $Y$, namely all the subsets of some fixed interval...