Math 199 (Assignments)

1. [To turn in on Wednesday]

   Let $\R^2$ be the set of all points $(x,y)$ in the $xy$-plane; i.e., ${\R^2}=\{\,(x,y)\mid x\in \R\hbox{ and }y \in \R\,\}$.

   Let $G=\{\,(x,y)\in{\R^2}\mid y=x^2\,\}$, the set of points $(x,y)$ in the plane where $y=x^2$, so $G$ is the graph of the function $f(x)=x^2$.

   Now let $X=\{\,(x,y)\in{\R^2}\mid \hbox{$(x,y)=(\cos t,1-\sin^2 t)$ for some $t\in\R$}\,\}$.

   1. Show that $X\subseteq G$. (Consider what we wrote at the end of class about how to prove the three types of statements.)
   2. Show that $G\not\subseteq X$. (Negate the condition for being a subset, and prove that.)
2. [Not to be turned in]

   Suppose $A$, $B$, and $C$ are sets. Prove that if $A\subseteq B$ and $B\subseteq C$ then $A\subseteq C$.

Math 199 (Fall 2019) web pages Created: 4 Oct 2019 Last modified: 7 Oct 2019 at 3:16 PM Comments to: dpvc@union.edu

Assignments