# $\sim$ is an equivalence relation

Note that $\sim$ as defined in the entry Landau notation is an equivalence relation on the set of all functions from $\mathbb{R}^{+}$ to $\mathbb{R}^{+}$. This set of functions will be denoted in this entry as $F$.

Reflexive (http://planetmath.org/Reflexive): For any $f\in F$, $\displaystyle\lim_{x\to\infty}\frac{f(x)}{f(x)}=1$, and $f\sim f$.

Symmetric: If $f,g\in F$ with $f\sim g$, then $\displaystyle\lim_{x\to\infty}\frac{f(x)}{g(x)}=1$. Thus:

$\begin{array}[]{ll}\displaystyle\lim_{x\to\infty}\frac{g(x)}{f(x)}&% \displaystyle=\lim_{x\to\infty}\frac{1}{\left(\frac{f(x)}{g(x)}\right)}\\ \\ &\displaystyle=\frac{1}{1}\\ \\ &=1\end{array}$

Therefore, $g\sim f$.

Transitive (http://planetmath.org/Transitive3): If $f,g,h\in F$ with $f\sim g$ and $g\sim h$, then $\displaystyle\lim_{x\to\infty}\frac{f(x)}{g(x)}=1$ and $\displaystyle\lim_{x\to\infty}\frac{g(x)}{h(x)}=1$. Thus:

$\begin{array}[]{ll}\displaystyle\lim_{x\to\infty}\frac{f(x)}{h(x)}&% \displaystyle=\lim_{x\to\infty}\left(\frac{f(x)}{g(x)}\cdot\frac{g(x)}{h(x)}% \right)\\ \\ &=1\cdot 1\\ \\ &=1\end{array}$

Therefore, $f\sim h$.

Title $\sim$ is an equivalence relation simIsAnEquivalenceRelation 2013-03-22 16:13:16 2013-03-22 16:13:16 Wkbj79 (1863) Wkbj79 (1863) 9 Wkbj79 (1863) Result msc 26A12