10.2.1. Summary of Proof Methods

Try these first

1. 1.

   What is a direct proof?

   Hint

   Think about what happens if P is true.

2. 2.

   If P is false, what can we say about P → Q?

   Hint

   Consider the definition of implications.

3. 3.

   What is a direct proof?

   • A method using contradiction
   • A method where P leads directly to Q
   • A method proving ¬P → Q

   Hint

   Think about the simplest way to prove an implication.

4. 4.

   If P is false, what can we conclude about P → Q?

   • True
   • False

   Hint

   Refer back to the definition of the implication.

5. 5.

   Prove that if a triangle has sides of lengths a, b, and c, and if a^2 + b^2 < c^2, then the triangle is not valid.

   Hint

   Start by assuming the triangle is valid and proceed to demonstrate inconsistency.

6. 6.

   Demonstrate using proof by contrapositive that if an integer n is even, then n^2 is even.

   Hint

   Work with definitions of odd and even integers to show results systematically.