4

What is the difference between the statements

$$ f(x) \neq 0 $$

and

$$ f(x) \not\equiv 0? $$

Jason Schmidt
  • 41

1 Answers1

5

$f(x)\neq0$ means when the $f$ maps the value $x$ to a non-zero value.

$f(x)\equiv0$ implies that $f(x)=0$ for all $x$. So $f(x)\not\equiv0$ means that there exists $x_0$, such that $f(x_0)\neq0$

QED
  • 12,644
  • (Both statements are equivalent.) – Pedro Dec 29 '13 at 06:33
  • 2
    Pedro Tamaroff is right if $x$ is not specified and is generic then the two are equivalent. However, if $x$ is known by the context of the statement, then Abishanka Saha is correct. It all depends on the context. It is like saying "for any $x$ there is a $y>x$" is same as "for all $x$ there is a $y>x$". Here "any" and "all" are equivalent – user44197 Dec 29 '13 at 06:37