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Mathematical Proof Techniques Quiz

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1. Which of the following statements is always true when using a direct proof to show that if $p$ then $q$?

2. What method of proof is typically used to show that a statement is false by assuming the opposite of what is to be proven?

3. Which proof technique is especially useful for proving statements that are formulated as 'for all $n$, if $n$ is a natural number, then...'?

4. To prove that 'if $p$ then $q$' using the contrapositive, you must show which of the following?

5. Which of the following is NOT a typical step in a mathematical induction proof?

6. What is the main purpose of using a proof by contrapositive?

7. In a proof by contradiction, why is it effective to assume the opposite of what you wish to prove?

8. When applying mathematical induction to prove propositions involving integers, what is the purpose of the 'inductive step'?

9. Which proof technique employs the structure 'if $p$ is true, then $q$ is true; if $p$ is false, then $q$ can still be true'?

10. In which scenario is proof by contradiction NOT a suitable method?