Determine the truth value of the statement: $(\forall x\in \mathbb{R},x^2\ge 0)\wedge (\exists x\in \mathbb{R},x^2<0)$.

Solution

1. The statement is a conjunction $P\wedge Q$ where $P:\forall x\in \mathbb{R},x^2\ge 0$ and $Q:\exists x\in \mathbb{R},x^2<0$.
2. $P$ is true because the square of any real number is non-negative.
3. $Q$ is false because there is no real number whose square is negative.
4. Since $P$ is true and $Q$ is false, $P\wedge Q$ is false.
5. Hence the answer is (B).