Quantifiers Math Example 4

Follow the full solution, then compare it with the other examples linked below.

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Determine the truth value of each and write its negation: (a)

\forall x \in \mathbb{R},\; x > 0

, (b)

\exists x \in \mathbb{Z},\; x^2 = 3

1
(a) False (e.g., $x = -1 \le 0$ ). Negation: $\exists x \in \mathbb{R},\; x \le 0$ (True).
2
(b) $x^2 = 3$ requires $x = \pm\sqrt{3}$ , which are irrational — not integers. False. Negation: $\forall x \in \mathbb{Z},\; x^2 \ne 3$ (True).

Answer

(a)\;\text{False; negation: }\exists x \le 0\;(\text{True}).\quad (b)\;\text{False; negation: }\forall x \in \mathbb{Z},\;x^2\ne 3\;(\text{True}).

The negation of a false statement is true, and vice versa. For quantified statements, switch

\forall \leftrightarrow \exists

and negate the predicate.

About Quantifiers

Symbols specifying the scope of a predicate: $\forall$ (for all, universal) and $\exists$ (there exists, existential).

Translate into symbols and determine the truth value: (a) 'Every natural number is positive.', (b) '

Negate the statement [formula] and determine the truth value of both the original and its negation.

Write in words: (a) [formula], (b) [formula].