in Set Theory & Algebra
905 views
0 votes
0 votes

A relation R on a set of positive integers is defined by (a,b) belongs to R iff a and b are relatively prime.

Which of the following is true about R?

a. Symmetric and Reflexive

b. Symmetric and irreflexive

c.Symmetric and transitive

d. Symmetric and  not transitive

 

The Ans is given as (d) but I think (b) is true. Any thoughts?

in Set Theory & Algebra
by
905 views

4 Comments

Two integers a and b are said to be relatively primemutually prime,or co-prime if the only positive integer (factor) that divides both of them is 1.

- wiki

This definition holds good with (1,1)

so the given relation can't be irreflexive

3
3
yeah got it thanks
0
0

Regarding is 1 is coprime of itself.

The numbers 1 and −1 are the only integers coprime to every integer, and they are the only integers that are coprime with 0.

Ref: https://en.wikipedia.org/wiki/Coprime_integers

0
0

1 Answer

0 votes
0 votes

Relation - is relatively prime.

Check Reflexive property:

aRa is not relatively prime . Hence not reflexive

Check Irreflexive property:

aRa

Ex. (4,8) is not relatively prime. Hence not Irreflexive

Check Symmetric property:

aRb => bRa

(3,4) is relatively prime and (4,3) also relatively prime. Hence Symmetric.

Check Transitive property:
aRb &bRc then aRc

Proof by Counter Example:

Ex:

(3,5) is relatively prime

(5,6) is relatively prime

but (3,6) is not relatively prime

Hence not transitive.

So option D is correct.

Related questions

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true