Quantifiers Math Example 1

Solution

1. 1
   The universal quantifier $\forall$ means 'for all'; the existential quantifier $\exists$ means 'there exists at least one.'
2. 2
   Translate: (a) 'Every natural number is positive' → $\forall n \in \mathbb{N},\; n > 0$. (b) 'There exists a real number $x$ such that $x^2 = 2$' → $\exists x \in \mathbb{R},\; x^2 = 2$.
3. 3
   Truth values: (a) True under the convention $\mathbb{N} = \{1,2,3,\ldots\}$ since all such $n \ge 1 > 0$. (If $0 \in \mathbb{N}$, the statement is False.) (b) True: $x = \sqrt{2} \in \mathbb{R}$ satisfies $(\sqrt{2})^2 = 2$.