If R is the smallest equivalence relation on the set \[\{1,2,3,4\}\] such that \[\{(1,2),(1,3)\} \subset \mathrm{R}\], then the number of elements in \[\mathrm{R}\] is \_\_\_\_\_\_\_\_\_\_.
Solutions · 1
Official solution
Since 1∼2 and 1∼3, all of 1,2,3 fall in the same equivalence class, while 4 remains its own class (nothing forces it to merge). The smallest equivalence relation is:
\[R = \{(1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)\}\]
(the diagonal, plus the full symmetric/transitive closure of {1,2,3} being mutually related, plus (4,4) alone).
Number of elements: 4+6=10