Jonathan Miles

Answered

2022-07-04

Suppose I have a string of inequalities, i.e.

$f(x,y,z)<g(x,y,z)<h(x,y,z)$

Clearly, if I find when $g(x,y,z)<h(x,y,z)$ and $f<g$, then $f<h$ when both these conditions hold.

Is it ever useful to compare $f(x,y,z)<h(x,y,z)$ in addition to $f<g$ and $g<h$

Or is it useful to compare the conditions (assuming I can get them in a comparable form) from $g<h$ and $f<h$ to each other? Maybe comparing the two can give a single, simplified condition for $f<g<h$ to hold?

$f(x,y,z)<g(x,y,z)<h(x,y,z)$

Clearly, if I find when $g(x,y,z)<h(x,y,z)$ and $f<g$, then $f<h$ when both these conditions hold.

Is it ever useful to compare $f(x,y,z)<h(x,y,z)$ in addition to $f<g$ and $g<h$

Or is it useful to compare the conditions (assuming I can get them in a comparable form) from $g<h$ and $f<h$ to each other? Maybe comparing the two can give a single, simplified condition for $f<g<h$ to hold?

Answer & Explanation

Jamiya Costa

Expert

2022-07-05Added 18 answers

Note that

$f(x,y,z)<g(x,y,z)<h(x,y,z)$

is equivalent to the following system

- $f(x,y,z)<g(x,y,z)$

- $g(x,y,z)<h(x,y,z)$

and the condition $f(x,y,z)<h(x,y,z)$ is implicitely assumed by the system.

$f(x,y,z)<g(x,y,z)<h(x,y,z)$

is equivalent to the following system

- $f(x,y,z)<g(x,y,z)$

- $g(x,y,z)<h(x,y,z)$

and the condition $f(x,y,z)<h(x,y,z)$ is implicitely assumed by the system.

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