2019Fall/MAT102: Introduction to Mathematical Proofs

This is not my first time teaching this course, but the level of excitement and nervousness doesn’t seem to go down at all at the beginning of this semester. I’m fortunate to have a whole class of students who engage fully during the lectures so far and make the teaching so enjoyable.

If you’re a current student enrolling in this course, or just want to take a look at what’s happening in class, you can find all the slides here:


This week we were talking about logic symbols and how quantifiers work, and how by changing the order of quantifies, we can tell completely different stories. Here’s one example we did in class:

 (\forall x \in \mathbb R)(\exists y \in \mathbb R)(y \geq x)

 (\exists y \in \mathbb R) (\forall x \in \mathbb R)  (y \geq x)

While the first statement is saying “for any real number x, we can always find a y such that  y \geq x ” which is a true statement, because we can simply make  y =x, the second statement is saying “we can find a real number y which is greater than or equal to all real numbers” which is a false statement. This is equivalent to say real numbers have an upper bound.

Can you tell the difference here?

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