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))).
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”.