Math 199 (Assignments)

First: Listen to the podcast for Monday's class

[Not to be turned in]

1. Suppose $A$ and $B$ are sets. Show that $A-B=\emptyset$ if, and only if, $A\subseteq B$. (Hint: consider using the contrapositive approach.)
2. Here we look at set difference more carefully.
   1. Prove that $A-(A-B)\subseteq B$.
   2. If $A$ and $B$ were numbers, the statement would be an equality. Find two specific sets $A$ and $B$ where the subset above is an equality.
   3. Show that equality need not hold by exhibiting two specific sets $A$ and $B$ where $A-(A-B)\ne B$.
   4. Determine a condition on $A$ and $B$ that would guarantee that the subset is actually an equality. Try to have your answer be as general as possible. The best solution will be one that is required for equality to hold.

      Prove that if your condition is true, then $A-(A-B)=B$.

      (Note that you already proved half of it in part (a), so you don't have to prove that again; just cite your earlier work.)

3. Suppose $a$, $b$, and $c$ are numbers. Prove that if $a\mid b$ and $b\mid c$ then $a\mid c$.

   (Remember, $p\mid q$ is about multiplication, not division, so you should not use division anywhere in your proof. Also, be careful of the "there exsits" statements that are part of the definition of $p\mid q$.)

Math 199 (Fall 2019) web pages Created: 14 Oct 2019 Last modified: 13 Oct 2019 at 7:51 PM Comments to: dpvc@union.edu

Assignments