###### tags: `108上助教` `數學`
# 數學導論HW6
## ch4-3

#### proof
$\mbox{Let }R = \{(x, y)\in \mathbb{R}\times\mathbb{R}|f(x) = f(y)\}\\
\mbox{We want to show that }R \mbox{ is a relation on }\mathbb{R}.$

$\mbox{Check }$
1. $\mbox{reflexive}$
$\mbox{Let } x \in \mathbb{R}\\
\because f(x) = x^2 = f(x)\\
\therefore (x, x)\in R$

2. $\mbox{symmetric}$
$\mbox{Let }x, y \in \mathbb{R} \mbox{ , and }(x, y) \in R\\
i.e. f(x) = f(y)\\
\Rightarrow x^2 = y^2\\
\Rightarrow y^2 = x^2\\
\Rightarrow f(y) = f(x)\\
\Rightarrow (y, x)\in R$

3. $\mbox{transitive}$
$\mbox{Let }x, y, z \in \mathbb{R} \mbox{ , and }(x, y),(y, z) \in R\\
\Rightarrow f(x)=f(y) , f(y) = f(z)\\
\Rightarrow x^2 = y^2 , y^2 = z^2\\
\Rightarrow x^2 = z^2\\
\Rightarrow f(x) = f(z)\\
\Rightarrow (x, z) \in R$

$\mbox{By 1. to 3. , we proved that } R \mbox{ is a relation on }\mathbb{R}$.

$[-7] = \{y \in \mathbb{R}|(-7, y)\in R\}$
$\mbox{That is, }\\
\mbox{find }y \mbox{ such that } f(-7) = f(y)\\
\Rightarrow \mbox{find }y \mbox{ such that } (-7)^2 = 49 = y^2\\
\Rightarrow y = 7 \mbox{ or } -7\\
\Rightarrow [-7] = \{7, -7\}\ \Box$