9.4. Rules of Inference for Quantified Statements

Try these first

1. 1.

   Translate the statement: 'All dogs bark.'

   Hint

   Remember the structure of universal statements.

2. 2.

   What does Existential Instantiation allow you to do?

   Hint

   Think about proving something for known cases.

3. 3.

   What does Universal Instantiation allow us to do?

   • Generalize statements
   • Conclude specific instances from general statements
   • Identify contradictions

   Hint

   Think about how we can apply rules in logic.

4. 4.

   True or False: Changing the order of quantifiers does not affect the meaning of statements.

   • True
   • False

   Hint

   Recall examples we discussed to validate this.

5. 5.

   Prove that for every even number n, n can be expressed as the sum of two prime numbers.

   Hint

   Explore established conjectures on primes.

6. 6.

   Using the rules discussed, prove the statement: ‘If any person is a teacher, they must have at least one student.’

   Hint

   Dissect the statements carefully.