The minimum number of elements that must be added to the relation R={(a,b),(b,c),(b,d)} on the set {a,b,c,d} so that it is an equivalence relation, is \_\_\_\_.
Answer: 13
Solutions · 1
Official solution
For R to become an equivalence relation on {a,b,c,d}, it must equal the full relation where a,b,c,d are all mutually equivalent (since a∼b, b∼c, b∼d links all four elements into one class):
\[\{(a,a),(b,b),(c,c),(d,d),(a,b),(b,a),(b,c),(c,b),(b,d),(d,b),(a,c),(a,d),(c,d),(d,c),(c,a),(d,a)\}\]
This has 16 elements. Since 3 are already given, the number that must be added is 16−3=13.