11.4. Proof Mechanisms for Existential Quantified Statements

Try these first

1. 1.

   Provide an example of a constructive proof.

   Hint

   Think of famous numbers related to sums of cubes.

2. 2.

   What is a uniqueness proof?

   Hint

   Connect this to the concept of solutions in equations.

3. 3.

   What type of proof gives a specific example to demonstrate existence?

   • Constructive Proof
   • Non-constructive Proof
   • Uniqueness Proof

   Hint

   Remember our discussions about 1729 and its properties.

4. 4.

   True or False: In a non-constructive proof, a typical example is provided.

   • True
   • False

   Hint

   Consider how we approached irrational numbers.

5. 5.

   Prove through both methods (constructive and non-constructive) that there exists a number that satisfies both rationality and irrationality properties, and discuss implications.

   Hint

   Consider both foundational properties and operations.

6. 6.

   Construct a uniqueness proof regarding the solutions of quadratic equations of the form ax^2 + bx + c = 0 and analyze scenarios where coefficients yield duplicate roots.

   Hint

   Evaluate the quadratic discriminant for insight on root duplication.