MAT102 Lecture 08B

• 04RelationsExercises_Students.pdf
• There are no positive integer solutions to $x^2=8y^2$
• Proof by Contradiction
  • Suppose there were two $x,y\in \mathbb{Z}$ which solve this equation
  • $x^2=8y^2$
  • $\frac{x^2}{8}=y^2$
  • $\frac{x}{\sqrt{8}}=y$
  • Unless we have $x=\sqrt{8},0$, $y$ will not be an integer.
  • Contradiction
• P is popular $P\subseteq \mathbb{R}$
• 1
  • Show that $⊢$ is transitive, but it is not reflexive or symmetric
  • To show transitivity we need that $a,b,c\in P$
  • If $a⊢b$ and $b⊢c$ then $a⊢c$
  • $a⊢b$
    • By definition, $a⊢b=b-a$
  • $b⊢c$
    • $c-b$
  • Then $a⊢c$
    • $c-a$
  • $c-a$
  • $c-a+b-b$
  • $c-b-a+b$
  • $c-b+b-a$
  • $(c-b)+(b-a)$
  • $(c-b)\in P$
  • $(b-a)\in P$
  • We're done
• 2
  • If $c\in P$ and $x⊢y$ then $cx⊢cy$
    • $c\in P$
    • so $\frac{1}{c}\in P$
    • $x⊢y$
      • $y-x$
    • Show that $cx⊢cy$
      • $cy-cx$
    • $y-x\in P$
    • $c(y-x)\in P$
    • $cy-cx\in P$
  • Show that if $c\notin P$ and is non-zero and $x⊢y$ then $cy⊢cx$
    • If $c\notin P$ then $-c\in P$
    • $x⊢y$
    • $y-x$
    • $-c(y-x)$
    • $-cy+cx$
    • $cx-cy\in P$
    • $cy⊢cx$
• 3
  • Show that if $x⊢y$ and $u⊢v$, then $x+u⊢y+v$
  • $x⊢y$
    • $y-x$
  • $u⊢v$
    • $v-u$
  • $x+u⊢y+v$
    • $(y+v)-(x+u)$
    • $y+v-x-u$
  • By $P_2$, we can have $x+y$ for any $x,y\in P$
  • $(y-x)+(v-u)$
  • $(y-x+v-u)$
  • $(y+v-x-u)$
  • $(y+v)-(x+u)\in P$
• 4
  • If $x,y\in P$ and $x⊢y$ show that $x^2⊢y^2$
  • By $P_2$ for every $x,y\in P$ we have that $xy\in P$
  • Let $x=x_1,y_1$ and let $y=x_2,y_2$
  • $x_1{y}_1⊢x_2{y}_2$
    • $x_2{y}_2-x_1{y}_1$
    • $y^2-x^2\in P$
  • $x^2⊢y^2$
• 5
  • If $x,y\in P$ and $x⊢y$ show that $y^{-1}⊢x^{-1}$
  • By $P_2$ for every $x,y\in P$ we have that $\frac{1}{x}\in P$
  • Let $x=\frac{1}{x}$ and $y=\frac{1}{y}$
  • $x⊢y$
    • $y-x$
    • $y^{-1}-x^{-1}$
    • $\frac{1}{y}-\frac{1}{x}$
    • $\frac{x}{xy}-\frac{y}{xy}$
    • $\frac{x-y}{xy}$
  • $y^{-1}⊢x^{-1}\in P$
    • $\frac{1}{x}-\frac{1}{y}$
• 6
  • This is the $<$ relation.