Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation R in the set *A* = {1, 2, 3, 4, 5, 6} as R = {(*x*, *y*): *y* is divisible by *x*}

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#### Solution

*A* = {1, 2, 3, 4, 5, 6}

R = {(*x*, *y*): *y* is divisible by *x*}

We know that any number (*x)* is divisible by itself.

=> (*x*, *x*) ∈R

∴R is reflexive.

Now,

(2, 4) ∈R [as 4 is divisible by 2]

But,

(4, 2) ∉ R. [as 2 is not divisible by 4]

∴R is not symmetric.

Let (*x*, *y*), (*y*, *z*) ∈ R. Then, *y* is divisible by *x* and *z* is divisible by *y*.

∴*z* is divisible by *x*.

⇒ (*x*, *z*) ∈R

∴R is transitive.

Hence, R is reflexive and transitive but not symmetric.

Concept: Types of Relations

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