11. Proof Strategies-II

Try these first

1. 1.

   Provide a counterexample for the universal statement: 'All integers are positive.'

   Hint

   Remember an integer can be negative.

2. 2.

   What does WLOG mean?

   Hint

   Think about simplifying cases in symmetrical arguments.

3. 3.

   What is a counterexample used for?

   • To prove a statement true
   • To disprove a statement
   • To illustrate a property

   Hint

   Think about proving versus disproving.

4. 4.

   True or False: A constructive proof provides no examples.

   • True
   • False

   Hint

   Recall what you understood about constructive proofs.

5. 5.

   Consider the statement: 'If a number is positive, then its square is positive.' Use a proof by cases to discuss how one would validate this without explicit value examples.

   Hint

   Explore both positive and negative possibilities in numeric realms.

6. 6.

   Demonstrate the non-constructive nature of proving that there are irrational numbers x and y such that xy is rational by using logical reasoning without establishing specific witnesses.

   Hint

   One valid path through exponentiation fulfilling conditions is your aim.