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

Answer 2

The relation of divisibility, in any ring with $1$, is an example of reflexive, transitive, but non symmetric relation

Answer 3

For this one time, I'll help you out... but indeed, next time maybe study the problem a bit more before asking for an answer ;-)

Take for A the set $A = \{ 1,2 \}$ and consider the relation ~ defined by 1 ~ 1, 2 ~ 2, and 1 ~ 2.

This relation satisfies a ~ a for any $a \in A$ (please check that!), but it does not satisfy 1) because 1 ~ 2 holds while 2~1 does not (this is because you should fill in the same element for `a' on both sides)...

Similarly, consider the same set but now with a relation ~ definded by 1 ~ 1, 2 ~ 2, 1~2 and 2~1. This relation does satisfy 1) but it does not satisfy 2) since 1~2 and 2~1 but 1 $\neq$ 2 (can you see why this is so?).