SUNY Geneseo Department of Mathematics
Friday, February 26
Math 239 03
Spring 2021
Prof. Doug Baldwin
(No.)
Next Tuesday (March 2) is the semester’s first rejuvenation day. It doesn’t affect this course directly, but in the spirit of getting a bit of a break from academic worries, I’m discouraging individual meetings that day (well, slightly discouraging them — if you want one you can certainly have it, but if you’d rather not, I think that’s a good idea in the spirit of the day). I also won’t hand out a new problem set today, so you’ll have a bit of a break from working on problem sets for the beginning of the week.
From “Beginning Activity 1 (An Introduction to Quantifiers)” and “Forms of Quantified Statements in English” in section 2.4 of the textbook and this discussion of quantifiers .
These mostly have good answers in the discussion, but it’s worth reviewing those answers and talking about why they’re correct.
Determine whether the following statements are true or false, and rewrite each using the quantifier symbols “∀” and “∃.”
Determine whether the following are true or false, and say why, and see if you can rephrase each in English, without quantifier symbols:
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 our classroom is writing on the whiteboard while we talk?
False, because there are no examples that make it true.
Is it true or false that every purple Martian in our classroom is writing on the whiteboard while we talk?
True. There are several ways to justify this:
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? Yes.
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?
(∀ x ∈ ∅)( P(x) ) is always true, for lack of counterexamples.
(∃ x ∈ ∅)( P(x) ) is always false, for lack of examples.
Negations of quantifiers (e.g., statements of the form ¬(∀ x ∈ S) P(x) or ¬(∃x ∈ S) P(x)) are tricky to understand at first, but are often important in understanding quantified statements. This is even more the case when the statement involves multiple quantifiers (which are hard enough to understand even without being negated). So we should study negations and statements with multiple quantifiers.
We explored a couple of examples, and it seemed that negations of existential statements involve universal quantifiers, and negations of universal statements involve existential quantifiers. To learn more about this…
Please read “Beginning Activity 2 (Attempting to Negate Quantified Statements),” “Negations of Quantified Statements,” “Counterexamples and Negations of Conditional Statements,” “Quantifiers in Definitions,” “Statements with More than One Quantifier,” and “Writing Guideline” in section 2.4 of our textbook.
Please also contribute to this discussion of advanced topics in quantifiers.