10.2. Conclusion

Try these first

1. 1.

   What is a direct proof?

   Hint

   Think about straightforward logical connections.

2. 2.

   What does it mean when we say a proof is vacuous?

   Hint

   Check the definition of vacuous.

3. 3.

   What is true about a direct proof?

   • It assumes the conclusion is false
   • It shows the premise leading to the conclusion
   • It uses indirect reasoning

   Hint

   Remember, direct proofs follow a straightforward logical path.

4. 4.

   True or False: A vacuous proof can be used when the premise is true.

   • True
   • False

   Hint

   Reflect on the conditions required for a vacuous proof.

5. 5.

   Prove using contradiction: If n is an odd integer, then n^2 must also be odd.

   Hint

   Focus on the implications of the assumption.

6. 6.

   Show that no positive integer squared can result in 2; use proof by contradiction.

   Hint

   Explore the properties of rational numbers and their divisibility.