GO Classes Set Theory And Algebra Practice Set 1 | Question: 1

Question

Let $A=\{0,1,2,3\}$ and $R$ a relation over $A$ :
$$
R=\{(0,0),(0,1),(0,3),(1,1),(1,0),(2,3),(3,3)\}
$$
Draw the directed graph of $R$. Check whether $R$ is an equivalence relation. Give a counterexample in each case in which the relation does not satisfy one of the properties of being an equivalence relation.

Answer 1

Given relation $R$ is an equivalence relation if:

1. it is reflexive
2. symmetric relation 
3. transitive relation

$R$ is not reflexive because $(2,2) \notin R$

$R$ is not symmetric relation as $(0,3)\in R, (3,0)\notin R,$similarly $(2,3)\in R, (3,2)\notin R$ 

$R$ is not transitive as $(1,0),(0,3)\in R ,(1,3)\notin R$

Answer 2

$R$ is not reflexive because $(2,2) \notin R$. It is not symmetric because $(3,2) \notin R$. It is not transitive because $(1,0)$ and $(0,3)$ are in $R$ but $(1,3) \notin R$.