Often one wants the reflexive-transitive closure of , which is the smallest transitive relation that contains and is also reflexive.
These can be defined explicitly as follows: if and only if, for some natural number , there exists a sequence such that
x = r_0 \sim \cdots \sim r_n = y .
If you accept as a natural number, then this defines the reflexive-transitive closure; if not, then this defines the transitive closure.
For the transitive closure, it's also possible to rephrase the above slightly (using only through ) to avoid any reference to equality.
In material set theory, the transitive closure of a pure set is the transitive set whose members are the elements of the downset of under the transitive closure (in the previous sense) of . That is, it consists of the members of , their members, their members, and so on.
Analogously, the reflexive-transitive closure of may be defined as the transitive closure of the successor of , which is the same as .