The free group "FS" with free generating set "S" can be constructed as follows. "S" is a set of symbols, and we suppose for every "s" in "S" there is a corresponding "inverse" symbol, "s"−1, in a set "S"−1. Let "T" = "S" ∪ "S"−1, and define a word in "S" to be any written product of elements of "T". That is, a word in "S" is an element of the monoid generated by "T". The empty word is the word with no symbols at all. For example, if "S" = {"a", "b", "c"}, then "T" = {"a", "a"−1, "b", "b"−1, "c", "c"−1}, and