1. The problem is to understand the concept of the universal quantifier in mathematics. 2. The universal quantifier is denoted by $\forall$ and means "for all" or "for every". 3. It is used to state that a property or condition holds for all elements in a given set. 4. Examples: - $\forall x \in \mathbb{R}, x^2 \geq 0$ means for every real number $x$, its square is non-negative. - $\forall x \in \mathbb{N}, x - 5 > 0$ means for every natural number $x$, $x$ minus 5 is greater than zero. - $\forall x, \forall y \in \mathbb{R}, x + y = y + x$ means for all real numbers $x$ and $y$, addition is commutative. 5. Important rules: - The statement must hold true for every element in the specified set. - The domain of the variable must be clearly specified. 6. In summary, the universal quantifier allows us to express properties that are true for all members of a set, which is fundamental in proofs and mathematical logic.