On relative ranks of full transformation semigroups

For a semigroup S and a set A subset of or equal to S the relative rank of S module A is the minimal cardinality of a set B such that A boolean OR B generates S. We show that the relative rank of an infinite full transformation semigroup module the symmetric group, and also module the set of all idempotent mappings, is equal to 2. We also characterise all pairs of mappings which, together with the symmetric group or the set of all idempotents, generate the full transformation semigroup.