Question

A binary relation $R$ on $N\times N$ is defined as follows: $(a,b)R(c,d)$ if $a\le c$ or $b\le d$. Consider the following propositions:

$P$ : $R$ is reflexive
$Q$ : $R$ is transitive

Which one of the following statements is TRUE

• A. Both $P$ and $Q$ are true
• B. $P$ is true and $Q$ are false
• C. $P$ is false and $Q$ are true
• D. Both $P$ and $Q$ are false

Answer 1

The correct option is B $P$ is true and $Q$ are false

$(a,b)R(c,d)$ if $a\le c$ or $b\le d$
$P:R$ is reflexive
$Q:R$ is transitive
Since, $(a,b)R(a,b)\Rightarrow a\le a$ or $b\le b\Rightarrow t$ or $t$
Which is always True. $R$ is reflexive.
Now let us check transitive property
Let $(a,b)R(c,d)\Rightarrow a\le c$ or $b\le d$
and $(c,d)R(e,f)\Rightarrow c\le e$ or $d\le f$
Now let us take a situation
$a\le c$ (True) or $b\le d$ (false)
and $c\le e$ (False) or $d\le f$ (True)
Now we can get neither $a\le e$ nor $b\le f$
So, $(a,b)R(c,d)$ and $(c,d)R(e,f)\text{/}\Rightarrow (a,b)R(e,f)$. So, clearly $R$ is not transitive.
So $P$ is true and $Q$ is false. Choice (b) is correct.