# DM - Functions & Relations -Q17

+1 vote

Let "Closure of a Relation R with respect to a Property P" be defined as follows:

let R be a relation on a set A. Then a relation R' is the closure of the relation R with respect to property P if and only if

1. R' satisfy the property P, and

2. , and

3. for any(every) relation R'' if and satisfies the property R, then

i.e if there is a relation R' with property P containing R such that R' is a subset of every relation R'' with property P containing R, then R' is called the closure of R with repect to P

Which of the following is False ?

(A). For any non-empty relation R on a non-empty set A, reflexive closure i.e. closure of R with respect to property “reflexivity”, can always exist.

(B). For any non-empty relation R on a non-empty set A, symmetric closure i.e. closure of R with respect to property “symmetricity”, can always exist.

(C). For any non-empty relation R on a non-empty set A, irreflexive closure i.e. closure of R with respect to property “irreflexivity”, can always exist.

(D). For any non-empty relation R on a non-empty set A, transitive closure i.e. closure of R with respect to property “transitivity”, can always exist. asked Jun 24
edited 4 days ago

+1 vote

Statement 1 :

For any non-empty relation R on a non-empty set A, reflexive closure i.e. closure of R with respect to property “reflexivity”, can always exist.

It is true. Reflexive closure of R can be found like this

=

Statement 2 :

For any non-empty relation R on a non-empty set A, symmetric closure i.e. closure of R with respect to property “symmetricity”, can always exist.

It is true. Symmetric closure of R can be found like this

=

Statement 3 :

For any non-empty relation R on a non-empty set A, irreflexive closure i.e. closure of R with respect to property “irreflexivity”, can always exist.

It is false. Consider this relation :

Now, we can not make it irreflexive by ADDING something to it.

Statement 4 :

For any non-empty relation R on a non-empty set A, transitive closure i.e. closure of R with respect to property “transitivity”, can always exist.

This is true.

https://en.wikipedia.org/wiki/Transitive_closure#:~:text=Informally%2C%20the%20transitive%20closure%20gives,337). answered 4 days ago by (112,390 points)