Quantifiers Math Example 2

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

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Negate the statement

\forall x \in \mathbb{R},\; x^2 \ge 0

and determine the truth value of both the original and its negation.

1
The original: $\forall x \in \mathbb{R},\; x^2 \ge 0$ . Since squares of real numbers are always non-negative, this is True.
2
Negate by switching $\forall$ to $\exists$ and negating the predicate: $\exists x \in \mathbb{R},\; x^2 < 0$ .
3
This says some real number has a negative square. Since no such real number exists, the negation is False.
4
As expected, the original and its negation have opposite truth values.

Answer

\text{Original: True.}\quad \neg(\forall x,\; x^2 \ge 0) = (\exists x,\; x^2 < 0):\;\text{False.}

The negation of

\forall x, P(x)

\exists x, \neg P(x)

. A statement and its negation always have opposite truth values, which provides a consistency check.

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) '

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

Determine the truth value of each and write its negation: (a) [formula], (b) [formula].