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

Answer 2

The given statement is a universally quantified statement: "For every real number \( x \), \( x^3 + 5 \) is positive". To negate a universally quantified statement, we change it to an existentially quantified statement and negate the predicate.

The negation is: "There exists at least one real number \( x \) such that \( x^3 + 5 \) is not positive".

This is equivalent to option (C). Option (A) is incorrect because it's possible for \( x^3 + 5 \) to be zero, which is not positive but also not negative. Option (B) is too broad; it includes cases where \( x^3 + 5 \) is negative and cases where \( x^3 + 5 \) is zero. Option (D) is simply the original statement.

Therefore, the correct negation is (C).