Express the following mathematical statement in predicate logic.

Question

Is this a correct expression of the mathematical statement, "Every positive real number has exactly two square roots.";

expression: ∀x∃a∃b((x>0) → (a!=b)∧(x=$a^2$)∧(x=$b^2$)).

Answer 1

The statement as written is true, but doesn't capture the word "exactly". Additionally, your quantification is a little odd: I'd have expected it to read $$(\forall x)(x>0 \to \exists a \exists b \dots)$$ because (while yours is true) it's a bit more work to check that yours really is true in the case that $x \leq 0$.

I'm going to use several different shapes of brackets to try to make this easier to read, but they're all just brackets. To capture "exactly", you want $$(\forall x)\left[x > 0 \to \left[(\exists a)(\exists b) \{(a \not = b) \wedge (a^2 = x) \wedge (b^2 = x) \color{red} {\wedge (\forall c)} \color{red} {\left((c^2 = x) \to [c = a \vee (c=b)] \right) }\} \right] \right]$$