[Math] If a relation is reflexive is it symmetric and transitive

Question

If a relation is reflexive is it symmetric and transitive ?

let ~ means " in relation with "

if A is a set , ~ is a relation on $A$, prove that:

if $a$~$a$ for any $a$ $\in$ A then

1- $x$~$y$ $\rightarrow$ $y$~$x$

2- $x$ ~$y$ , $y$ ~ $z$ $\rightarrow$ x~z

if this is wrong , give an example to a reflexive relation which is not transitive or symmetric

Answer 1

Let $A=\{a,b,c,d,e\}$ and $$R=\{(a,a),(b,b),(c,c),(d,d),(e,e),(c,e),(e,b)\}$$