Give an example of Deductive Reasoning and a example of a counterexample.

Deductive reasoning: By definition it is when you move from a general statement, such as "All men are mortals", to a more specific statement as "Socrates is a man", through a logical thought process as in "Therefore, Socrates is mortal".

Other example:

"This dog always barks when someone is at the door, and the dog didn't bark. " Conclusion: There's no one at the door.

Counterexample: is an example with a negative connotation. Whereas an example may be used to support or illustrate a claim, a counterexample is used to refute an assertion.

Example:

The assumption that every English word contains at least one vowel. Which is simply not true. One of the more exotic counterexamples is the word nth - of a mathematical origin.