Question

Negation of the statement $\forall x \in R, x^2+1=0$ is

• A. $\exists x \in R \text{ such that } x^2+1 < 0$
• B. $\exists x \in R \text{ such that } x^2+1 \neq 0$
• C. $\exists x \in R \text{ such that } x^2+1 \le 0$
• D. $\exists x \in R \text{ such that } x^2+1 = 0$

Hint:

When negating quantified statements, remember to flip both the quantifier ($\forall \leftrightarrow \exists$) AND the equality/inequality symbol ($= \leftrightarrow \neq$, $< \leftrightarrow \ge$, $> \leftrightarrow \le$).

Answer 1

The Correct Option is B

Step 1: Flip the quantifier.
The negation of "for all" is "there exists".

Step 2: Flip the inside.
The inside condition $x^2 + 1 = 0$ becomes $x^2 + 1 \neq 0$.

Step 3: Put together.
So the negation reads: there exists an $x$ in R with $x^2 + 1 \neq 0$.
\[ \boxed{\exists x \in R \text{ such that } x^2 + 1 \neq 0} \]