10. Relations

Relations

Binary relation

Binary relation

For sets $A$ and $B$, a (binary) relation from $A$ to $B$ is any subset $R$ of the [[Cartesian product of sets|Cartesian product]] $A \times B$ or otherwise a set consisting of some ordered pairs.

We use notation $\boxed{aRb}$ to denote that $(a,b)\in R$, and say that $a$ is $R$-related to $b$. If $(a,b)\notin R$, then we right $\boxed{a\cancel{R}b}$.

Link to original

For example, let $P$ be the set of people, and $C$ be the set of Among Us crewmates. Define a relation $\text{Suspicious\_of}$ by taking

$$\text{Suspicious\_of}=\{(x,y)\in P\times C|x\text{ is suspicious of }y\}$$

If I am suspicious of Red but think Blue is a crewmate, then

$$(\text{insert, Red})\in \text{Suspicious\_of}\text{ but }(\text{insert, Blue})\notin \text{Suspicious\_of}$$

This can also be written as

$$\text{insert Suspicious\_of Red}\text{ and }\text{insert }\cancel{\text{Suspicious\_of}}\text{ Blue}$$

Relations on a set

Relation on a set

A relation from a set $A$ to $A$ itself is called a relation on $A$.

It can also be said that:

• a relation on a set $A$ is a subset of $A\times A$
• a relation on a set $A$ is a set consisting some ordered pairs of elements from $A$

Link to original

For example, let $A$ be the set of all people in the class.

$$\begin{array}{rl}{R}_1& =\{(u,v)\in A\times A|u\text{ likes }v\}\\ R_2& =\{(u,v)\in A\times A|u\text{ is taller than }v\}\end{array}$$

Will is 180cm tall and likes Jill who is 165cm. But Jill does not like Will. Hence, $(\text{Will, Jill})\in R_1$, $(\text{Will, Jill})\in R_2$, $(\text{Jill, Will})\in R_1$ and $(\text{Jill, Will})\in R_2$.

Relations on the set $Z$ of integers

• ‘smaller than’: $<=\{(x,y)\in \mathbb{Z}\times \mathbb{Z}|x\text{ is smaller than }y\}$
  • $5$ is smaller than $25$, so $(5,25)\in <$ or more commonly $5<25$
  • $10$ is not smaller than $2$ so $(10,2)\notin <$ or more commonly $10\nless 2$
• ‘smaller than or equal to’: $\le =\{(x,y)\in \mathbb{Z}\times \mathbb{Z}|x\text{ is smaller than or equal to }y\}$
  • $-1\le -1$
  • $1\nleqq 0$
  • $<\subset \le$
• ‘divisibility’: $\text{div}=\{(x,y)\in \mathbb{Z}\times \mathbb{Z}|x\text{ divides }y\}$
  • $(1,25)\in \text{div}$
  • $(6,3)\notin \text{div}$