An integer m is said to be related to another integer n if m is a integral multiple of n. This relation in Z is reflexive, symmetric and transitive.

Question

State True or False for the statement:

An integer m is said to be related to another integer n if m is a integral multiple of n. This relation in Z is reflexive, symmetric and transitive.

Answer 1

False

Let R be the relation defined on Z by mRn if m is a integral multiple of n.

Let mRm ∈ R

⇒ m is a integral multiple of m.

Which is true since m is integral multiple of itself.

Thus, R is reflexive.

Let mRn ∈ R

⇒ m is a integral multiple of n

⇒ m= zn ∀ z ∈ Z

⇒ n is not integral multiple of m.

⇒ nRm ∉ R

Thus, R is not symmetric.

Let mRn ∈ R and nRp ∈ R

⇒ m is a integral multiple of n and n is a integral multiple of p

⇒ m is a integral multiple of p

⇒ mRp ∈ R

Thus, R is transitive.

Hence, the given statement is false.