Write the negation of the following statement (in words):

Question

"For any field $F$, and any $a\in F$, if $a^3 = 1$ then $a = 1$."

Is this statement TRUE OR FALSE?

Is the negation TRUE OR FALSE?

Attempt:

There is a field $F$ and there is an $a \in F$ such that if $a^3 ≠ 1$ then $a ≠ 1$.

Answer 1

Not quite: What you need for the negation is

"There is a field $F$ and there is an $a\in F$ such that $a^3 = 1$ and $a\neq 1$."

That is, the negation of $a^3 = 1 \rightarrow a = 1$ is $$\lnot(a^3 = 1 \rightarrow a = 1)\equiv \lnot(a^3 \neq 1 \lor a = 1) \equiv a^3 = 1 \land a \neq 1$$