MAT511 homework,         due Oct 22, 2003

1. Let $A$ and $B$ be nonempty sets. Prove that $A\times B = B\times A$ if and only if $A=B$. What if one of $A$ or $B$ is empty?

2. For each of the relations below, indicate whether it is reflexive, symmetric, or transitive. Justify your answer.
   1. $\le$ on the set ${\mathbb{N}}$.
   2. $\perp = \left\{{ (l,m) \,\,\vrule{}\,\,\mbox{$l$\ and $m$\ are lines, with $l$ perpendicular to $m$\ }}\right\}$.
   3. $\sim$ on ${\mathbb{R}}\times{\mathbb{R}}$, where $(x,y) \sim (z,w)$ if $x+z \le y+w$.
   4. $\smile$ on ${\mathbb{R}}\times{\mathbb{R}}$, where $(x,y) \smile (z,w)$ if $x+y \le z+w$.
   5. $\square$ on ${\mathbb{R}}\times{\mathbb{R}}$, where $(x,y) \square (z,w)$ if $x+z = y+w$.

3. Prove that if $R$ is a symmetric, transitive relation on a set $A$, and the domain of $R$ is $A$, then $R$ is reflexive on $A$.

4. Consider the relations $\sim$ and $\square$ on ${\mathbb{N}}$ defined by $x \sim y$ iff $x+y$ is even, and $x \square y$ iff $x+y$ is a multiple of 3. Prove that $\sim$ is an equivalence relation, and that $\square$ is not.

5. For each $a \in {\mathbb{R}}$, let $P_a = \left\{{(x,y) \in {\mathbb{R}}\times{\mathbb{R}}\,\,\vrule{}\,\,y = a - x^2}\right\}$.
   1. Sketch the graph of $P_{-2}$, $P_{0}$, and $P_{1}$.
   2. Prove that $\left\{{P_a \,\,\vrule{}\,\,a \in {\mathbb{R}}}\right\}$ forms a partition of ${\mathbb{R}}\times{\mathbb{R}}$.
   3. Describe the equivalence relation associated with this partition.