11.2. Proof by Cases

Try these first

1. 1.

   Give a counterexample for the statement 'All integers are positive.'

   Hint

   Think of integers that are less than zero.

2. 2.

   What does WLOG stand for?

   Hint

   It allows you to simplify your proof.

3. 3.

   What is proof by cases?

   • A single example
   • Breaking down a statement
   • A generalization

   Hint

   Think about how we validate universal statements.

4. 4.

   True or False: A counterexample can prove a universally quantified statement is true.

   • True
   • False

   Hint

   Remember what a counterexample represents.

5. 5.

   Prove that for any integer n, if n is odd, then n² is also odd using proof by cases.

   Hint

   Break down what odd means in terms of definition.

6. 6.

   Discuss the implications of constructing a non-constructive proof regarding irrationality and rationality.

   Hint

   Explore examples like √2 raised to itself.