Properties of given binary relation?

Question

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?

1. Both $P$ and $Q$ are true.
2. $P$ is true and $Q$ is false.
3. $P$ is false and $Q$ is true.
4. Both $P$ and $Q$ are false.

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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?

Answer 1

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...

Answer 2

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.