When attempting to negate mathematical statements I struggle to know what part of the sentence I need to negate and what I can leave. For example, inequalities and $\in$
As general rules I know If I have a predicate $P(x)$ and a sentence such as $\forall x: P(x)$ I should change $\forall$ to $\exists$ and negate the predicate. Similarly if it was $\exists x: P(x)$ I would change $\exists$ to $\forall$ and that I should negate the predicate.
As an example I took the following from Rudin's Principles of Mathematical Analysis.
Prior to negating I write this as: $\forall \{t_{n}\} \in (x,b) \wedge t_{n} \rightarrow x \Rightarrow f(t_{n}) \rightarrow q$
After negating I believe that the answer should be: $\exists \{t_{n}\} \in (x,b) \wedge t_{n} \rightarrow x : f(t_{n}) \not\rightarrow q$
Question:
In the above would the predicate not be $\{t_{n}\} \in (x,b) \wedge t_{n} \rightarrow x$ so that this should be negated if I follow the general rule I listed above, but I would then get an incorrect negation?
Best Answer
You made a small but important mistake in translating this to symbols. The actual statement would be better written as $$ \forall {t_n} \in (x,b) : (t_n\rightarrow x\Rightarrow f(t_n)\rightarrow q)$$ As you can see from the parentheses I added, the quantifier is outside the implication. To negate the whole sentence, you change $\forall$ to $\exists$ then negate the implication, which results in $$ \exists {t_n} \in (x,b) : (t_n\rightarrow x\wedge f(t_n) \nrightarrow q)$$ If you’re confused about negating an implication, remember that $A\Rightarrow B$ is equivalent to $B\vee\neg A$.