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: Read the statement.
The statement is: for all $x\in R$, $x^2+1=0$. We want its negation.
Step 2: Recall the rule for negating for all.
The negation of for all ($\forall$) is there exists ($\exists$).
Step 3: Recall the rule for negating the condition.
The inner condition flips to its opposite, so $x^2+1=0$ becomes $x^2+1\neq 0$.
Step 4: Put both changes together.
The negation reads: there exists $x\in R$ such that $x^2+1\neq 0$.
Step 5: Sanity check.
Indeed for every real $x$, $x^2+1$ is at least $1$, never $0$. So the negation is the true statement, which fits.
Step 6: Match the option.
This is option (2). \[ \boxed{\exists\, x\in R \text{ such that } x^2+1\neq 0} \]