Homogenous relations An equivalence relation must be Reflexive , Transitive , Graph of a relation . A relation is given by its graph A partition of a relation Exercise 1.
R=x\ r\ y \Leftrightarrow x\lt y\\ S=x\ s\ y \Leftrightarrow x/y\\ T=x\ t\ y \Leftrightarrow gcd(x,y)=1\\ V=x\ v\ y \Leftrightarrow x\equiv y\pmod 3 \end{align}$$ Write the graphs R,S,T,V $$R=\{(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)\}$$$$S=\{(2,4),(2,2),(2,6),(3,6),(3,3),(4,4),(5,5),(6,6)$$ $$T=\{(2,3),(2,5), (3,4),(3,5),(4,5),(5,6),(3,2),(5,2),(5,3),(5,6),(6,5)\}$$ $$V=\{(5,2),(2,5),(6,3),(3,6),(3,3),(2,2),(4,4),(5,5),(6,6)\}$$ Exercise 2. $$\begin{gathered} |A|=n\in\mathbb{N}^*\\ |B|=m\in\mathbb{N}^*\\ \text{Find the no. of }\\ i. \varphi: A\rightarrow B (2^{nm})\\ ii. \varphi: A\rightarrow A (2^{n^2})\\ \text{the formula is: } 2^{|Domain\times codomain|} \end{gathered}$$ Exercise 4. $$\begin{gathered} (\mathbb{R},\lt)\rightarrow T\\ (\mathbb{N}, |)\rightarrow R,T\\ (\mathbb{Z},|)\rightarrow R,T\\ (\mathbb{R}^2,\perp)\rightarrow S\\ (\mathbb{R}^3,\parallel)\rightarrow R,T,S\\ (\mathbb{R}^3,\equiv_\Delta)\rightarrow R,T,S\\ (\mathbb{R}^3, \backsimeq_\Delta)\rightarrow R,T,S \end{gathered}$$ Exercise 5. $$\begin{gathered} M=\{1,2,3,4\}\\ r_1,r_2 - \text{relations}\\ \pi_2,\pi_2 -\text{partitions}\\ R_1=\Delta_M\cup\{(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)\}- R,S,T\Rightarrow \text{equiv}\\ R_2=\Delta_M\cup\{(1,2),(1,3)\}- \Rightarrow\text{not equiv}\\ \pi_1=\{\{1\},\{2\},\{3,4\}\}- \text{yes}\\ \pi_2=\{\{1\},\{1,2\},\{3,4\}\}- \{1\}\cup\{1,2\}=\emptyset, \text{ no }\\ i. r_1,r_2, relations\\ ii. \pi_2,\pi_2 partitions \end {gathered}$$ Exercise 6. $$\begin{gathered} (\mathbb{C},r),(\mathbb{C},s)\\ z_1\ r\ z_2 \Leftrightarrow |z_1|=|z_2|\\ z_1\ s\ z_2\Leftrightarrow arg\ z_1= arg\ z_2\text{ or } z_1=z_2=0\\ \\ i. r,s- equiv.\ R,T,S\Rightarrow \text{equiv.}\\ ii. \mathbb{C}_{/r}\ , \mathbb{C}_{/s}\\ \\ \mathbb{C}_{/r}=\{x\in \mathbb{C}| \ |x|=y\}\\ \\ s - R, T, S \Rightarrow \text{equiv.} \\ \mathbb{C}_{/s}=\{x\in\mathbb{C}| arg\ x=y \text{ or } x=0\} \end{gathered}$$ Exercise 7. $$\begin{gathered} n\in\mathbb{N}\\ (\mathbb{Z},\rho_n)\\ x\ \rho_n\ y\Leftrightarrow n|(x-y)\\ \\ 1. \rho_n - \text{equiv.}\\ 2. \mathbb{Z}/\rho_n=? (n=0,n=1)\\ \\ \begin{align} \forall x,y,z\in\mathbb{Z}, x\ \rho_n\ y, z\ \rho_n\ y\\ x\ \rho_n\ y= &n |(x-y)\\ y\ \rho_n\ z= &n|(y-z)\\ &+\\ &n|(x-z)\\ \end{align}\\ n|(x-z)\\ \forall x,y\in \mathbb{Z}, x\rho_n y, y\rho_nx?\\ x\ \rho_n\ y \end{gathered}$$ TBC^ $$\begin{gathered}\mathbb{Z}\rho_n=\{x\in\mathbb{Z}|n|(x-y),y\in\mathbb{Z}\}\\ =\{x\in\mathbb{Z}|2|(x-y),y\in\mathbb{Z}\}=\{x,y\in\mathbb{Z}|x\equiv y\pmod 2\}\\ =\{\hat0, \hat1\}=\mathbb{Z}_2\\ \mathbb{Z}/\rho_n=\mathbb{Z}_n=\{\hat0,\hat1,\ldots,\hat{n-1}\}\\ n=0\Rightarrow \mathbb{Z}/\rho_0=\emptyset\\ x\rho_0 y\Rightarrow0|(x-y) \text{Not}\\ n=1\Rightarrow\mathbb{Z}/\rho_1=\mathbb{Z}\\ x\rho_1y\Rightarrow 1|(x-y) \end{gathered}$$ >[!info] What is a function > >$|h<x>|=1,\forall x\in M$ > Translation : $\exists !y\in M\text{ s.t. } h(x)=y$ Exercise 9. - [ ] 🔼 write problem sentence $$\begin{align} h<x>&=\{y\in M|h(x)=y\}\\ &=\{y\in M|x=4k+y, k\in \mathbb{Z}\}\\ &=\{y\in M|x\equiv y\pmod 4\}\\ x=1\implies y=1\text{ unique}\\ x=2\implies y=2\text{ unique}\\ \ldots M=\{0,1,2,3\}\implies \exists!\in M\text{ s.t. } h(x)=y\\ \implies |h<x>|=1\\ \implies h - \text{function}\ \end{align}