10.1.3.3. Proof by Contradiction

Try these first

1. 1.

   What does it mean to prove something by contradiction?

   Hint

   Think about negation!

2. 2.

   Is the statement 'If 5 is even, then 10 is greater than 5' vacuously true?

   Hint

   Focus on the truth value of the premise.

3. 3.

   What is the main purpose of proof by contradiction?

   Hint

   Remember the process of negation leading to invalidity.

4. 4.

   True or False: Proof by contrapositive shows that p → q is equivalent to ¬q → ¬p.

   Hint

   Think about logical statements.

5. 5.

   Prove that for any integer n, if n^2 is odd, then n must be odd using contradiction.

   Hint

   Use the properties of even and odd integers.

6. 6.

   Show that if n is a natural number such that n^2 < n, then n must be 1.

   Hint

   Consider properties of inequalities with natural numbers.