1. ## Smallest equivalence relation

Find the smallest equivalence relation R on M = {1; 2; 3; 4; 5} which contains the subset Ro = {(1; 1); (1; 2); (2; 4); (3; 5)} and give its equivalence classes.

How would I go about doing this ?

Any help would be good

cheers

2. ## Re: Smallest equivalence relation

To make it reflexive, it needs (2,2), (3,3), (4,4), (5,5)
To make it transitive: (1,2) and (2,4) needs (1,4).
To make it symmetric, it needs (2,1), (4,2), (4,1), (5,3)

Equivalence classes: [1]=[2]=[4]={1,2,4}, [3]=[5]={3,5}

3. ## Re: Smallest equivalence relation

Thanks slip eternal ...when it says smallest equivalence ration are we just basically breaking it down to the smallest amount of sets possible?
Thanks

4. ## Re: Smallest equivalence relation

Originally Posted by bee77
Thanks slip eternal ...when it says smallest equivalence ration are we just basically breaking it down to the smallest amount of sets possible?
Thanks
When it says smallest, it means with the minimum number of pairs.

5. ## Re: Smallest equivalence relation

Ah Thanks SlipEternal

6. ## Re: Smallest equivalence relation

Originally Posted by bee77
Find the smallest equivalence relation R on M = {1; 2; 3; 4; 5} which contains the subset Ro = {(1; 1); (1; 2); (2; 4); (3; 5)} and give its equivalence classes.
How would I go about doing this ?
Originally Posted by SlipEternal
Equivalence classes: [1]=[2]=[4]={1,2,4}, [3]=[5]={3,5}
Just a note on this process.
SlipEternal gave you the real key to your question "How would I go about doing this ?"
$\mathscr{R}=[\{1,2,4\} \times\{1,2,4\}]\cup[\{3,5\}\times\{3,5\}]~.$

As point of interest, there is a one-to-one relationship between partitions of a set and equivalence relations on that set.

7. ## Re: Smallest equivalence relation

Thanks for the help...not sure if i suck or my lecturer does ..but you guys explain it better,cheers