Question

The negation of the statement “For every real number $x$, $x^2 + 5$ is positive” is:

• A. For every real number $x$, $x^2 + 5$ is not positive
• B. For every real number $x$, $x^2 + 5$ is negative
• C. There exists at least one real number $x$ such that $x^2 + 5$ is not positive
• D. There exists at least one real number $x$ such that $x^2 + 5$ is positive

Answer 1

The Correct Option is C

1. Understand the problem:

We need to find the negation of the statement:

“For every real number \(x\); \(x^2 + 5\) is positive.”

2. Recall the negation of universal quantifiers:

The negation of “For all \(x\), \(P(x)\)” is “There exists an \(x\) such that not \(P(x)\).”

3. Apply to the given statement:

The original statement is:

\[ \forall x \in \mathbb{R}, \; x^2 + 5 > 0 \]

Its negation is:

\[ \exists x \in \mathbb{R}, \; x^2 + 5 \leq 0 \]

In words: “There exists at least one real number \(x\) such that \(x^2 + 5\) is not positive.”

Correct Answer: (C) There exists at least one real number \(x\) such that \(x^2 + 5\) is not positive