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?


For reflexivity, since the condition is an "OR", you're happy with proving that one of the two conditions holds - although in this case both hold true. And you have to consider a generic pair $(a, b)$.
For transitivity, think well. You have a condition either on the first components or on the second component. And this time around they need not both hold.
Spoiler

Think of the three pairs $(2,2), (2, 1), (1,1)$. You have $(2, 2) R (2, 1)$ and $(2, 1) R (1, 1)$, but...

