Chapter 1.6, Problem 4E

Provide either a proof or a counterexample for each of these statements.

1. For all positive integers $x,x^2+x+41$ is a prime.
2. $\left(\forall x\right)\left(\exists y\right)\left(x+y=0\right).$ (Universe of all reals)
3. $\left(\forall x\right)\left(\forall y\right)\left(x>1\wedge y>0\Rightarrow y^x>x\right).$ (Universe of all reals)
4. For integers a, b, c, if a divides bc, then either a divides b or a divides c.
5. For integers a, b, c, and d, if a divides $b-c$ and a divides $c-d,$ then a divides $b-d.$
6. For all positive real numbers $x,x^2-x\ge 0.$
7. For all positive real numbers $x,2^x>x+1.$
8. For every positive real number x, there is a positive real number y less than x with the property that for all positive real numbers $z,yz\ge z.$
9. For every positive real number x, there is a positive real number y with the property that if $y<x,$ then for all positive real numbers $z,yz\ge z.$