# 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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### What is Math Comm

MAA Mathematical Communication (mathcomm.org) is a developing collection of resources for engaging students in writing and speaking about mathematics. The site originated in the MIT Department of Mathematics and was expanded through support from an NSF grant.