# How is symmetry in an inequality determined? I was reading a book about inequalities, in that I found that a/b+b/c+c/a>1 is a symmetric inequality in a,b,c. But if I change the order from (a,b,c) to (a,c,b), I am not getting the same inequality which is the basic definition of symmetric inequality. What am I doing wrong?

How is symmetry in an inequality determined?
I was reading a book about inequalities, in that I found that
$\begin{array}{}\text{(1)}& \frac{a}{b}+\frac{b}{c}+\frac{c}{a}>1\end{array}$
is a symmetric inequality in a,b,c. But if I change the order from (a,b,c) to (a,c,b), I am not getting the same inequality which is the basic definition of symmetric inequality. What am I doing wrong?
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Zara Pratt
It's not symmetric inequality because the permutation $\left(a,b,c\right)\to \left(a,c,b\right)$ gives another inequality:
$\frac{a}{c}+\frac{b}{a}+\frac{c}{b}>1.$
By the way, the inequality
$a+b+c>abc$
is symmetric.