Can "such that" be treated as "and" to form a single sentence? $f$ being any real valued function, $k1, k2$ being two positive scalars. The scenario is:

*

*"do something" if:
$$x-f(x)<k_1 ~~~~s.t.~~~|x|>k_2$$
Can I rewrite it as:

*

*"do something" if
$$x-f(x)<k_1~~~AND~~~ |x|>k_2$$
$$\implies \frac{x-f(x)}{|x|}<\frac{k_1}{k_2}$$
 A: 

*

*"do something" if $$x-f(x)<k_1 ~~~~s.t.~~~|x|>k_2$$

Perhaps the author means this:

*

*"do something" if $$|x|>k_2\implies x-f(x)<k_1.$$

Can I rewrite the scenario as:

*

*"do something" if $$x-f(x)<k_1~~~AND~~~ |x|>k_2$$ $$\implies \frac{x-f(x)}{|x|}<\frac{k_1}{k_2}$$

Given that $``D \text{ if } Q"$ and that $\color{cyan}P⟹Q$ and $Q⟹\color{violet}R,$ then:

*

*these are meaningful but invalid

*

*$D\text{ if }\color{violet}R\color\red{\quad\quad\longleftarrow\text{You meant this.}}$

*$D\text{ if }(Q⟹\color{violet}R)\color\red{\quad\quad\longleftarrow\text{You wrote this; it isn't equivalent to the previous line.}}$

*$D\text{ if }(Q\text{ implies }\color{violet}R)\color\red{\quad\quad\longleftarrow\text{You wrote this.}}$



*these are valid

*

*$D \text{ if }\color{cyan}P$

*$(D \text{ if }Q);\text{ therefore }(D \text{ if }\color{cyan}P)$

*$(D \text{ if }Q),\text{ which implies }(D \text{ if }\color{cyan}P)$



*this is also valid but conveys the least information as it does not claim that $(D \text{ if }Q)$ is actually true

*

*$(D \text{ if }Q)\text{ implies }(D \text{ if }\color{cyan}P).$
A: "Such that" is only used paired with an "There exists". The condition should be
"do something" if:
$$\text{there exists}~~x~~\text{ such that}~~~ x-f(x)<k_1 ~~~~\text{AND}~~~|x|>k_2$$
