# Example: Examples of Relations

(related to Definition: Relation)

### Relations defined between elements of the same set

• Let $P$ be the set of all people. The silbing relation $$S\subset P\times P$$ is the subset of all pairs $(p_1,p_2)$ of persons such that $S=\{(p_1,p_2):a \text{ is a sibling of } b\}.$
• Let $C$ be the set of all cities in a country. We define the (ternary) relation $R\subset C\times C\times C$ as the subset of all tripples of cities $(c_1,c_2,c_3)$ such that there is a railway connection between $c_1$ to $c_3$ via $c_2.$
• Let $\mathbb N$ be the set of natural numbers. We define the order relation $"\ge"\subset \mathbb N\times \mathbb N$ as all pairs of natural numbers $(a,b)$ such that $a-b\in\mathbb N.$ In this case we write $a\ge b$.
• Let $T$ be the set of all triangles in a plane. We define the congruency relation $=$ between these triangles as all pairs of triangles $(t_1,t_2)$ such that $t_1$ can exactly cover $t_2$ by moving or rotating it in the plane. In this case we write $t_1=t_2$ and say that these triangles are congruent.
• Order relation of integers $$R_3=\{(x,y)\in\mathbb Z\times \mathbb Z\mid x \le y\}.$$
• Multiples of $4$ $$R_4=\{(x,y)\in\mathbb Z\times \mathbb Z\mid y=4x\}.$$
• Let $U$ be the set of all users of a social network. The "is a follower of" relation is defined $$R_5=\{(u,v)\in U\times U\mid u\text{ is a follower of }v\}.$$
• A relation can also be defined for only one set: Let $U$ be the set of all users of a social network. The "is a bot" relation is defined $$R_6=\{(u)\in U\mid u\text{ is a robot }\}.$$
• ...

### Relations defined between elements of different sets

• Let $P$ be the set of all people. The relation of all people $P$ who have the same child is $$R_1=\{(a,b)\in P\times P\mid a,b\text{ are the parents of the same child}\}.$$
• Let $U$ be the set of all Internet users and $V$ be the set of all visitors of a specific website. $$R_2=\{(u,v)\in U\times V\mid u\text{ is female},v\text{ visited the website in the last 2 days}\}.$$
• Let $L$ be the set of all lectures and $T$ be the set of all lecturers in a university. We define a binary relation $C\subset L\times T$ as consisting of all pairs of a lecture $\lambda\in L$ and a lecturer $l\in T$ such that the lecturer $l$ gives the lecture $\lambda.$
• Let $S$ be the set of all students of the same university. We define another relation $B\subset S\times L$ such that a student $s\in S$ attends the lecture $l\in L$.
• Let $C$ be the set of all countries in the world and let $\mathbb R$ be the set of real numbers. If we map each country $c$ to its area $a$ (measured in square kilometers), be define a binary relation of pairs $(c,a)\in C\times\mathbb R.$

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