Introductory Quantifiers Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

The quantifiers “for all” and “there exists,” or their synonyms, are small but important parts of many mathematical statements. This discussion helps you start getting familiar with them and their meanings.

Quantifiers and Their Meanings

See if you can determine whether the following statements are true or false. Also try to rewrite each using the quantifier symbols “∀” and “∃.”

1. For all natural numbers n, n > 0.
2. Every natural number is greater than -1.
3. There exists an integer n such that n > 0.
4. Some integer is negative.
5. If y is a real number, then ln y > 0.
6. Sometimes ln y > 0, where y is a real number.
7. One or more of the integers is less than 10.

Determine whether the following are true or false, and see if you can rephrase each in English, without quantifier symbols:

1. (∀ x ∈ ℝ)(x ∈ ℚ)
2. (∃ x ∈ ℝ)(x ∈ ℚ)
3. (∃ n ∈ ℤ)(n > 8 ∧ n < 5)
4. (∀ m ∈ ℤ)(m > 8 ∨ m < 10)

Empty Sets

Some of the most unintuitive quantified statements involve quantification over an empty set (i.e., statements of the form “there exists an x in S such that …” or “for all x in S, …” where S is an empty set.

For example, is it true or false that some purple Martian in your room is helping you answer this question?

Is it true or false that every purple Martian in your room is helping you answer this question?

Years ago, one of my students in another class that was learning about quantifiers came to me with this story: one evening my student, who had never in his life taken a physics course, noticed that his roommate was studying for a physics exam. He walked over to the roommate and casually remarked “just so you know, I got an A in every physics course I ever took.” Was my student telling the truth?

Can you state a general rule for the truth value of statements that are universally quantified over an empty set? What about for statements that are existentially quantified over an empty set?