1. The problem is to negate the proposition: $$\forall x \in \mathbb{R}, \exists y \in \mathbb{R} : x^2 + y^2 = 1$$ 2. The original statement says: "For every real number $x$, there exists a real number $y$ such that $x^2 + y^2 = 1$." 3. To negate a statement of the form $\forall x, \exists y : P(x,y)$, we use the logical equivalence: $$\neg (\forall x, \exists y : P(x,y)) \equiv \exists x, \forall y : \neg P(x,y)$$ 4. Applying this to our proposition, the negation is: $$\exists x \in \mathbb{R}, \forall y \in \mathbb{R} : x^2 + y^2 \neq 1$$ 5. In words, this means: "There exists a real number $x$ such that for every real number $y$, the equation $x^2 + y^2 = 1$ does not hold." This is the negation of the original proposition.