The question asks to analyze the proposition $\forall x (x^2 \geq x)$, where the domain for $x$ is the set of positive real numbers, and to determine which statement correctly describes this proposition.

Proposition breakdown:

• The expression $x^2 \geq x$ must hold for all $x \in \mathbb{R}^+$ (all positive real numbers).
• We need to test this inequality to see if it's true for all values or if there are any counterexamples.

Step 1: Simplifying the inequality

The inequality can be rewritten as: $x^2 \geq x$ Dividing both sides by $x$ (since $x > 0$, division is valid): $x \geq 1$ Thus, the inequality $x^2 \geq x$ holds if and only if $x \geq 1$.

Step 2: Checking counterexamples

• For $x = 1$, we have $1^2 = 1$, which satisfies the inequality.
• For $x = \frac{1}{2}$, we have $\left( \frac{1}{2} \right)^2 = \frac{1}{4}$, and $\frac{1}{4} < \frac{1}{2}$, so the inequality does not hold.

Thus, $x = \frac{1}{2}$ serves as a counterexample where the proposition is false.

Correct answer:

Since the proposition is false and $x = \frac{1}{2}$ is a counterexample, Option C is correct:

• C: The proposition is false, and $x = \frac{1}{2}$ is a counterexample.

Would you like more details or have any further questions?

Here are five related questions to deepen your understanding:

1. What happens if $x$ includes negative real numbers? How would that affect the inequality $x^2 \geq x$?
2. How does the behavior of quadratic functions influence inequalities like $x^2 \geq x$?
3. What would the proposition look like if $x$ were restricted to integers?
4. What is the relationship between counterexamples and the validity of universal quantifiers like $\forall x$?
5. How do we solve inequalities that involve powers of $x$?

Tip: When dealing with inequalities and universal quantifiers, always check for edge cases and counterexamples to confirm the proposition's truth value.