- Write the statement in the language of the logic of predicates, the condition of the equivalence of two equations $f_1(x)=0$, $f_2(x)=0$
$$\forall x(((x \in \mathbb{R})∧(f_{1}(x)=0)∧(f_{2}(x)=0))→(f_{1}(x)=f_{2}(x)))$$ - is it wrong?
- Define the truth domain of a predicate
$$P(x) = \forall y \exists z ((y \in \mathbb{N} ) \to ((z \in \mathbb{R} ) \land (y^{x} < y^{z})))$$
You have said that at any value, $x$, if the two functions equal zero, they equal each other. Which is trivially true, but not what you were required to say.
You were asked to make an assertion of equivalence of the equations. First step is to ask, "What does this mean?"
Equivalent equations are those which have exactly the same solutions. That is, if any value, $x$, is the solution to the equation, $f_1(x)=0$, then it is also the solution to the equation, $f_2(x)=0$ too, and vice versa.
Second step: Say this with symbols.
Do that thing.