Order of quantifiers

Context: This lesson plan is from a weekly communication recitation that accompanies M.I.T.’s Real Analysis (18.100C). This week students learn about differentiability and the mean value theorem (Rudin pp. 103-110). This recitation revisits concepts that were introduced in the second recitation.

Authors: The recitation was developed primarily by Joel B. Lewis and Craig Desjardins.

Communication objectives: Translating among mathematical concepts, mathematical language, and notation; with particular attention to how changing the order of quantifiers changes the meaning.

Students worked in small groups on the following task:

Give some examples of f: R–>R having each of the following properties.

  • A function f: X–>Y is said to have property P1 if for all ε > 0 and for all x1, x2 in X there exists δ > 0 such that if d1(x1, x2) < δ, we have that d2(f(x1), f(x2)) < ε.
  • A function f: X–>Y is said to have property P2 if for all ε > 0 there exists δ > 0 such that for all x1, x2 in X with d1(x1, x2) < δ, we have that d2(f(x1), f(x2)) < ε.
  • A function f: X–>Y is said to have property P3 if there exists δ > 0 such that for all ε > 0 and for all x1, x2 in X with d1(x1, x2) < δ, we have that d2(f(x1), f(x2)) < ε.

Then, as class, we discussed the meaning of each of the statements above. The following points came up in discussion:

  • Note that f is continuous on its domain if for all x1 in X and for all ε > 0 there exists δ > 0 such that for all x2 in X with d1(x1, x2) < δ, we have that d2(f(x1), f(x2)) < ε. [This is not one of the listed properties.]
  • Note that P2 is uniform continuity, which can be illustrated by drawing windows on a graph.
  • P1 is satisfied by every function.
  • On the reals, the only functions satisfying P3 are constant functions, but this is not true in every metric space.

Note: The above activity does not fill an entire recitation. In Fall 2009 the activity was combined with the conclusion of peer critique that had been started in the preceding recitation.

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