Final answer:
The proposition ∀x(x2≥x) is false, and x=1/2 is a counterexample.
Step-by-step explanation:
The proposition ∀x(x2≥x) states that for every positive real number x, the square of x is greater than or equal to x. To determine if this proposition is true or false, we can evaluate it using counterexamples. A counterexample is an example that disproves a proposition.
Let's evaluate the proposition for x=1. Plugging in x=1, we get 12 ≥ 1, which simplifies to 1 ≥ 1. Since this inequality is true, x=1 is not a counterexample.
Now, let's evaluate the proposition for x=1/2. Plugging in x=1/2, we get (1/2)2 ≥ 1/2, which simplifies to 1/4 ≥ 1/2. Since this inequality is false, x=1/2 is a counterexample to the proposition.
Therefore, the proposition ∀x(x2≥x) is false, and x=1/2 is a counterexample.
The complete question is:
The domain for variable x is the set of positive real numbers. Select the statement that correctly describes the proposition ∀x(x2≥x). The proposition is false, and x=1 is a counterexample. The proposition is false, and x=−1 is a counterexample. The proposition is false, and x=1/2 is a counterexample. The proposition is true.