11.5. Proof of Uniqueness

Try these first

1. 1.

   What is a uniqueness proof?

   Hint

   Think of examples where only one solution works.

2. 2.

   What does a counterexample demonstrate?

   Hint

   Recall a false mathematical statement you've seen.

3. 3.

   What is a uniqueness proof?

   • A proof showing many solutions
   • A proof showing there is exactly one solution
   • A proof that disproves a statement

   Hint

   Think about how many examples would satisfy the condition.

4. 4.

   True or False: A counterexample proves a statement can be true.

   • True
   • False

   Hint

   Consider what it means to prove something 'wrong'.

5. 5.

   Prove that if x is an irrational number, then x² is also irrational.

   Hint

   Think about the definition of rationality.

6. 6.

   In a system of equations, show that for certain coefficients, a unique solution exists.

   Hint

   Recall the properties of matrices and determinants.