For each of the following, either prove the identity or provide a counterexample to show the identity is not always true.
- For all real , .
- For all real , there exists a real such that .
- For every real , there exists a real such that .
- There exists a real such that .
- False.
Counterexample: Let , then , and . Hence, we have , but ; thus, the statement is false. - False.
Counterexample: Let , then , but . - True.
Proof. Let , then for all , and for all - True.
Proof. Let , then