- #1

- 679

- 30

##f(x)=\begin{cases}\frac{1}{x^2-1},&x\leq1\\ \frac{1}{1-x^2},&x>1\end{cases}##

If it's true, would you say it is non-vacuously true for all values of ##x## except ##1## and vacuously true for the case ##x=1##? Can a statement be both non-vacuously true and vacuously true at the same time?

A conditional statement with a false antecedent is vacuously true. But is the existence of ##f(x)## part of the antecedent?