Question

On the set N of all natural numbers define the relation R by aRb if and only if the Greatest common divisor(GCD) of a and b is 2. Then R is
A. Reflexive, but not symmetric
B. Symmetric only
C. Reflexive and transitive
D. Reflexive, symmetric and transitive

Answer 1

Hint: Here, we will use the definitions of reflexive, symmetric and transitive relations to check whether the given relations are reflexive, symmetric or transitive.

Complete step-by-step answer:
A relation between two sets is a collection of ordered pairs containing one object from each set. If the object x is from the first set and the object y is from the second set, then the objects are said to be related if the ordered pair (x, y) is in the relation.
It is given that R is a relation defined by aRb if and only if the Greatest common divisor(GCD) of a and b is 2. We will check if it is reflexive, symmetric or transitive or not.
For a reflexive relation, aRa should exist. That is, GCD of a and a should be 2, which is not true because GCD of two equal numbers is the number itself. Hence, R is not reflexive.

For a symmetric relation, both aRb and bRa should exist. That is, if GCD of a and b is 2, the GCD of b and a is also 2, which is true. Hence, R is symmetric.

For a transitive relation, if aRb and bRc exist, aRc should also exist. That is if GCD of a and b is 2, and GCD of b and c is also 2, then GCD of a and c should be 2. This is not necessarily true. For example, GCD of 4 and 6 is 2, GCD of 6 and 8 is 2 but the GCD of 4 and 8 is 4. So, it is not transitive.

R is symmetric only. The correct option is B. Symmetric only

Note: In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.Students should note here that for a relation to be a particular type of relation, i.e. reflexive, symmetric or transitive, all its elements must satisfy the required conditions. If we are able to find even a single counter example, i.e. if any of the elements doesn’t satisfy the criteria, then we can’t proceed further.