# a ~ b Relations

• Nov 7th 2012, 02:55 AM
Matt1993
a ~ b Relations
I am really struddling to understand these types of questions. Please can someone walkthough it with me.

The Question is

On the set N, define a relation ~ by stipulating that a ~ b if
hcf(a,b) = 1 (where a; b 2 belong to N). Decide if this relation is (a) reflexive, (b) symmetric and
(c) transitive, in each case giving either a proof or a counterexample, as appropriate.

Any help would be great thanks
• Nov 7th 2012, 04:02 AM
Matt1993
Re: a ~ b Relations
I think i have managed to do part (a)
if a~a then hcf(a,a) = a. If a =1,2,3,4,5.... the hcf(a,a)=1,2,3,4,5... Not 1 for all cases. Therefore not reflexive
• Nov 7th 2012, 04:13 AM
Plato
Re: a ~ b Relations
Quote:

Originally Posted by Matt1993
The Question is
On the set N, define a relation ~ by stipulating that a ~ b if
hcf(a,b) = 1 (where a; b 2 belong to N). Decide if this relation is (a) reflexive, (b) symmetric and (c) transitive, in each case giving either a proof or a counterexample, as appropriate.

You know that
\$\displaystyle 8\sim 9~~\&~~9\sim 10\$ so what does that tell you about transitivity?
• Nov 7th 2012, 04:43 AM
Matt1993
Re: a ~ b Relations
that its not transative because hcf(8,9)=1 and hcf(9,10)=1 but hcf(8,10)= 2
• Nov 7th 2012, 05:17 AM
Plato
Re: a ~ b Relations
Quote:

Originally Posted by Matt1993
that its not transative because hcf(8,9)=1 and hcf(9,10)=1 but hcf(8,10)= 2

That is correct. The relation is symmetric. WHY?
• Nov 7th 2012, 05:24 AM
Matt1993
Re: a ~ b Relations
is it because hcf(a,b) = hcf(b,a) = 1. a and b are both still coprime