Advanced 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.)

This discussion helps you become familiar with some ideas about quantifiers beyond their basic meanings. In particular, it explores statements that involve multiple quantifiers, how to negate statements with quantifiers, and how to prove quantified statements.

Quantified Statements and Their Negations

Try to write each of the following statements using the quantifier symbols “∃” and “∀” instead of English words. For statements that involve the word “not,” try to find two symbolic forms, one that uses the logical “not” symbol (“¬”) and one that doesn’t:

1. For every natural number there is some greater natural number.
2. It’s not the case that for every natural number there is some smaller natural number.
3. For every pair of real numbers x and y there is some other real number between x and y.
4. Some negative integer is larger than all other negative integers (you might want set builder notation along with quantifiers for this; it’s a little tricky in another way too).
5. Some integer is even (build the definition of “even,” in terms of quantifiers, into the answer).
6. It’s not true that every integer is even (again, include a quantifier-based definition of “even” in the answers).

Proofs

Can you describe a general strategy for proving an existentially quantified statement (i.e., a “there exists” statement). For example, how would you prove “there exists an integer greater than 10”? Does the method you used generalize to other existential statements?

How about a universally quantified statement (i.e., a “for all” statement)? For example, how would you prove “every even integer is of the form n + 2 where n is another even integer”? Does your method generalize to other universal statements?

(And as a sort of “bonus” question, how many quantifiers are there in that “every even integer is of the form n + 2 where n is another even integer” example?)