Question

A relation R is defined on a set of N natural numbers such that $aRb$ if $a$ is a divisor of $b$ then R is a partial order relation not a total order relation.
(A) True
(B) False

Answer 1

Hint: We solve this problem by first going through the definitions of partial order relation and total order relation. Then we check if the given relation is reflexive, antisymmetric and transitive thereby proving that it is a partial order relation. Then we check if it satisfies the conditions for total order relation or not and find the whether the given statement is true.

Complete step-by-step answer:
First let us go through the definition of partial order relation and total order relation.
A relation R on set S is said to be a partial order relation if it follows three properties
i) Reflexive, For all $a\in S$, $aRa$.
ii) Anti-Symmetry, If $aRb$ and $bRa$ then $a=b$.
iii) Transitive, If $aRb$ and $bRc$ then $aRc$.

A relation R is said to be a Total order relation if it follows the properties
i) It is a Partial order relation
ii) For any $a,b\in S$, either $aRb$ or $bRa$.

Now let us consider our given relation R on set N such that $aRb$ if $a$ is a divisor of $b$.
First let us check if the relation is reflexive, that is for all $a\in S$, $aRa$.
For any $a\in N$, a is always a divisor of a. So, $aRa$ is true. So, R is reflexive
Now let us check if it is anti-symmetric, that is if $aRb$ and $bRa$ then $a=b$.
$aRb$ means a is a divisor of b and $bRa$ means b is a divisor of a.
For any two numbers, a is a divisor of b and b is a divisor of a if and only if a=b.
So, R is anti-symmetric.
Now let us check if R is transitive, that is if $aRb$ and $bRc$ then $aRc$.
$aRb$ means a is a divisor of b, that is b=ka
$bRc$ means b is a divisor of c, that is c=lb
Substituting value of b in c, c=lb=lka.
So, a is a divisor of c. So, we get $aRc$. So, R is transitive.
As R follows all the three properties, it is a partial order relation.
Now let us see if it is a total order relation or not.
For R to be a total order relation it need to satisfy the property, for any $a,b\in S$, either $aRb$ or $bRa$.
Now let us consider two numbers from the set N say 5 and 7.
As we see them, 5 is not a divisor of 7 and 7 is not a divisor of 5. So, R does not satisfy the above property. So, it is not a total order relation.
Hence, we get that R is a partial order relation and not a total order relation.
Hence the given statement is true.
Hence the answer is Option A.

Note: he common mistake one makes while solving this problem is one might take the property in the partial order relation as symmetric, that is one might take the definition as
A relation R on set S is said to be a partial order relation if it follows three properties
i) Reflexive, For all $a\in S$, $aRa$.
ii) Symmetry, If $aRb$ then $bRa$.
iii) Transitive, If $aRb$ and $bRc$ then $aRc$.