11.1. Disproving Universally Quantified Statements

Try these first

1. 1.

   Define a counterexample in the context of universally quantified statements.

   Hint

   Think about why one example is enough to disprove an assertion.

2. 2.

   What does 'proof by cases' entail?

   Hint

   Consider whether all potential scenarios have been addressed.

3. 3.

   What is a counterexample in the context of disproving a universally quantified statement?

   • A statement that supports the universal claim.
   • An example that showcases a case where the universal claim does not hold.
   • A proof of the universal claim.

   Hint

   Think about scenarios where a universal claim fails.

4. 4.

   True or False: Proof by cases means you only need to prove one scenario.

   • True
   • False

   Hint

   Consider how many cases you must consider.

5. 5.

   Explore whether the statement 'All prime numbers are odd' is valid. Provide a counterexample, and analyze the implications of proving such a statement false.

   Hint

   Look for the simplest prime numbers.

6. 6.

   Demonstrate how proof by cases could simplify proving the inequality a² > b² for positive integers. Break down the analysis into relevant cases.

   Hint

   Think about expressions that exhibit these properties.