Algebra

Relations & Functions

Medium

core

Let a relation R = {(a, b) : a is a factor of b, a, b $\in$ N}. Then, R is ______.

Reflexive and Symmetric but NOT Transitive.

Symmetric and Transitive but NOT Reflexive.

Reflexive and Transitive but NOT Symmetric.

Equivalence Relation.

✅ Correct Option: 3

The relation R = {(a, b) : a is a factor of b, a, b $\in$ 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 $\in$ N.

If b divides c, then c = b × m for some m $\in$ 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.

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Let a relation R = {(a, b) : a is a factor of b, a, b ∈\in∈ N}. Then, R is ______.