The relation R = {(a, b) : a is a factor of b, a, b ∈ N} means (a, b) is in R when a is a factor of b.
For example: (2, 6) is in R because 2 is a factor of 6.
For reflexivity, check if every number is a factor of itself.
Is a a factor of a?
Since a = a × 1, every natural number is a factor of itself.
Therefore, R is reflexive.
For symmetry, check if a is a factor of b implies b is a factor of a.
Consider (2, 6) in R, where 2 is a factor of 6.
However, 6 is not a factor of 2.
Therefore, R is not symmetric.
For transitivity, check if a is a factor of b and b is a factor of c implies a is a factor of c.
If a divides b, then b = a × k for some k ∈ N.
If b divides c, then c = b × m for some m ∈ N.
Substituting:
c = (a × k) × m
c = a × (km)
This means a divides c.
Therefore, R is transitive.
R is reflexive and transitive but not symmetric.