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
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
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}
Just a note on this process.
SlipEternal gave you the real key to your question "How would I go about doing this ?"
Once the Equivalence classes are identified the your answer comes:
$\mathscr{R}=[\{1,2,4\} \times\{1,2,4\}]\cup[\{3,5\}\times\{3,5\}]~.$
As point of interest, there is a one-to-one relationship between partitions of a set and equivalence relations on that set.