Wednesday 30 September 2015
Having learnt direct proofs last week, we are going to study how we can make such proofs with quantified statements.
1.1 Universal Generalisation
Related reading: Stein et al. p. 180–181 or Rosen p. 72–73
Universal generalisation is the straight-forward direct proof of a universally quantified statements. This is essentially similar to the conditional proof, to prove an implication.
Problem 1.1 Consider the following assertation which we want to prove:
The sum of two odd integers is even.
- Rephrase the statement symbolically as an implication.
- Prove the implication.
- Rephrase the statement symbolically as a quantified statements, making all quantifiers explicit.
- Prove the quantified statement.
Exercise 1.1 Consider the assertion that if is odd, then is odd.
- Formulate the assertion symbolically as an implication.
- Prove the resulting statement using a conditional proof.
- Rephrase the assertion with explicit quantification.
- Prove the resulting statement using universal generalisation.
Exercise 1.2 Are there any implicit quantifiers in the statement, the product of odd integers is odd? Which?
Exercise 1.3 Prove that the product of two odd integers is odd.
1.2 Analysing quantified expressions
Related reading: Stein et al. p. 168–173 or Rosen p. 41–45
Theorem 1 The following two statements are equivalent:
Exercise 1.4 Consider the following slight modification of the theorem:
Is this statement true? Either give a counter-example or an argument to justify it.
Exercise 1.5 Consider the following slight modification of the theorem:
Is this statement true? Either give a counter-example or an argument to justify it.
Problem 1.2 Which of the following statements are true and which are false?
The following definitions are used
This video presents a summary of the different principles we have used for direct proofs.
Related reading: Stein et al. p. 181–182 or Rosen p. 62 and 72