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: Understanding the Question:
We are asked to find the logical negation of a universally quantified mathematical statement.

Step 2: Key Formula or Approach:
The negation of a universal quantifier "for all" ($\forall$) is the existential quantifier "there exists" ($\exists$).
Furthermore, the negation of the condition $P(x)$ becomes its logical opposite, $\sim P(x)$.
Mathematically: $\sim(\forall x, P(x)) \equiv \exists x, \sim P(x)$.

Step 3: Detailed Explanation:
The original statement is: $\forall x \in R, x^2+1=0$.
Applying the negation rule:
1. The quantifier $\forall x \in R$ transforms into $\exists x \in R$.
2. The condition $x^2+1=0$ transforms into $x^2+1 \neq 0$.
Combining these gives the negated statement: $\exists x \in R \text{ such that } x^2+1 \neq 0$.

Step 4: Final Answer:
The negation is $\exists x \in R \text{ such that } x^2+1 \neq 0$, which matches option (B).