Session 2
3.1 Quantifiers
Related reading: Stein et al. p. 163–172 or Rosen p. 34–41
Predicate logic (or first order logic) generalises propositional logic by introducing variables into the predicates.
Exercise 3.1 Consider the predicate
For what values of is true …
- if the universe of is the integers (i.e. )?
- if the universe of is the set of real numbers (i.e. )?
Definition 3 A quantifier is an expression or operator which turns a statement about an arbitrary element into a statement about a universe.
The expression there is some is an existential quantifier. Mathematically we write for this quantifier, and means «there exists some such that is true».
Exercise 3.2 (Video «The existential quantifier») An equation can be thought of as a predicate. Consider the equation .
Express the claim that the equation has a real solution (a solution ) in symbolic form.
Besides the existential quantifier , we have the universal quantifier . The expression means that «for any , is true».
Exercise 3.3 Consider the two statements
- It was raining every day throughout our holiday.
- There was a rain-free day during our holiday.
Define predicate symbols and formulate the expression in symbolic form using quantifiers.
Can you see some relationship between the statements? (Implication? Equivalence? Other?)
Exercise 3.4 A natural number is prime if it has no positive integer factor other than one and itself. Find two ways to write this criterion in symbolic form. You need to introduce a variable to denote potential factors of , and use either the existential or universal quantifier on for two different ways.
Each answer could begin with the words « is a prime number if and only if ...».
Exercise 3.5 Which of the following statements are true and which are false?
The following definitions are used
Exercise 3.6 What is the difference between the two following statements?
Discuss. Is there a difference at all?