Properties of given binary relation? A binary relation R on $N×N$ is defined as follows$: (a,b)R(c,d)$ if $a≤c$ or $b≤d$. Consider the following
propositions:
$P: R$ is reflexive
$Q: R$ is transitive
Which one of the following statements is TRUE?


*

*Both $P$ and $Q$ are true. 

*$P$ is true and $Q$ is false. 

*$P$ is false and $Q$ is true. 

*Both $P$ and $Q$ are false.




My attempt :
Reflexive$: (a, a) R(a, a)$
Since $a \leq a$, or $a \leq a$
Transitive$: (a, b) R (c, d)$ or $(c, d) R(m, n)$ then $(a, b) R(m, n)$
Suppose $(a, b) R(c, d)$
$\implies a \leq c$ or $b \leq d$
and $(c, d) R(m, n)$
$\implies c \leq m, d \leq n$
Since $a \leq c$, or $c \leq m$ so $a \leq m$
$b \leq d$ or $d \leq n$, so $b \leq n$
$\implies (a, b) R(m, n)$

Can you explain in formal way, please?


Reflexive: $ (a, b) R (a, b)$
That is true, since $a \leq a$, or $b \leq b$ (actually both inequalities are true)     
Transitive:
$(20,40) R (30,30)$ is true by definition
$(30,30) R (10,50)$ is true by definition              
But it's not true that  $(20,40) R (10,50)$.  
In other words: from ($a \leq a_1$, or $b \leq b_1$) and ($a_1 \leq a_2$, or $b_1 \leq b_2$),
it does not follow that ($a \leq a_2$, or $b \leq b_2$). 
So $R$ is not transitive.   
