Question

1. Discuss the following relations for reflexivity, symmetricity and transitivity:
i) The relation R defined on the set of all positive integers by "mRn if m divides n".
(ii) Let P denote the set of all straight lines in a plane. The relation R defined by "l R m if l is perpendicular to m"
​iii) let A be the set consisting of all the members of a family. The relation R defined by "aRb if a is not a sister of b "

Answer 1

Answer:

1)

The relation R defined on the set of all positive integers by "mRn if m divides n".

We know that

Every integer divides itself so the relation is reflexive

Since

3 divides 6 but 6 does not divides 3 so the relation is not symmetric

And

If a divides b and b divides c then a also divides c   where a , b and c is the positive integer so the relation is transitive.

2)

Let P denote the set of all straight lines in a plane. The relation R defined by "l R m if l is perpendicular to m"

Since the line L is not perpendicular to itself so the relation is not reflexive.

If the line L is perpendicular to line M then the line M is also perpendicular to line L to the relation is symmetric.

If line L is perpendicular to line M and M is perpendicular to line F then Line L and Line F are parallel to each other so the relation is not transitive.

3)

let A be the set consisting of all the members of a family. The relation R defined by "aRb if a is not a sister of b "

Since the sister S is note the sister of itself so the relation is reflexive.

If the A is not the sister of B then B is also not the sister of A so the relation is symmetric.

And

If A is not the sister of B and B is not the sister of C Then it is possible that the A and C are sisters of each other So the relation is not transitive.

If A , B and C each have different parents then the relation is transitive.