Chapter 0.2, Problem 32E

Find a proof or exhibit a counterexample to each of the following statements.

1. $2x^2+3y^2>0$ for all real numbers $x$ and $y$.

2. $a$ an even integer $\to$ $\frac{1}{2}a$ an even integer.

3. For each real number $x$, there exists a real number $y$ such that $xy=1$.

4. If $a$ and $b$ are real numbers with $a+b$ rational, then $a$ and $b$ are rational.

5. If $a$ and $b$ real numbers with $ab$ rational $\to$ $a$ and $b$ rational.

6. $b^2-4ac>0$ and $a\ne 0\to p(x)=a{x}^2+bx+c$ has two distinct real roots.

7. $x^2\ge x$ for all real numbers $x$.

8. $n^2\ge n$ for all positive integers $n$.