I gave my math students a Surprise Formative Assessment today. This was a quick check of their retention of a definition, and facility at forming the negation of a statement.

Everyone did okay on the definition – it’s important to know precisely what things mean.

The results of the negation exercise showed three areas that are slowing our progress through the IBL notes. I identify them here, and I’d love for my students to discuss them with each other in study groups, on the subway, in office hours, etc.

**First,** expressing and identifying universal and existential quantifiers in informal but precise English usage. Many statements we use in mathematics state that some property is ALWAYS TRUE for EVERY object in a class of objects. These are universally quantified statements. Others state that THERE EXISTS at least one object from a class of objects for which some property holds. These are existentially quantified statements. It must be clear from your mathematical writing which sort of statement you mean!

**Second,** the skill of correctly NEGATING a quantified statement. If a statement is NOT ALWAYS true, then THERE EXISTS an object for which the statement is false; a counterexample. Likewise, if THERE DOES NOT EXIST an object about which a statement is true, then the statement is false FOR EVERY OBJECT.

**Third,** in a statement that involves multiple and nested quantifiers, there is something interesting and useful that happens when you negate it. We saw this today when negating the statement that “Sequence p_1, p_2, … converges to point x”, which is triply-quantified. Here it is, with parentheses to emphasize the scope of the quantifiers.

*FOR ALL open intervals S that include x, (THERE EXISTS a positive integer N, so that (FOR ALL integers n that are at least N, (p_n is in S.)))*

This has the format *FOR ALL(THERE EXISTS(FOR ALL(statement)))*. When you negate it, it becomes: *THERE EXISTS(FOR ALL(THERE EXISTS(negation of statement)))*.

That is,

*THERE EXISTS an open interval S that includes x for which (THERE DOES NOT EXIST a positive integer N, for which (FOR ALL integers n that are at least N, (p_n is in S.)))*

We can move the negation further across the quantifiers:

*THERE EXISTS an open interval S that includes x so that (WHENEVER N is a positive integer, then it’s not true that (FOR ALL integers n that are at least N, (p_n is in S.)))*

pushing the negation along even further:

*THERE EXISTS an open interval S that includes x so that (WHENEVER N is a positive integer, THERE EXISTS some integer n that’s at least as big as N, for which (p_n is NOT in S.)))*

This last form is the most useful when crafting a proof by contradiction. If you assume that the sequence p_1, p_2, … does not converge to the point x, then you can immediately assume this special open interval S with some amazing special properties.

For further background on negating statements, I recommend “The Nuts and Bolts of Proof” by Antonella Cupillari, which has a nice section entitled “How to Construct the Negation of a Statement”.

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