Consider G = ⟨c, d | c^2 = d^3⟩. A = ⟨c | c^2⟩ & B = ⟨d | d^3⟩.
As a transversal for A take Y = {1, c} and for B take Z = {1, d, d^−1}.
Compute the normal forms of the elements
c^3 d^−2 c^−4 d^4 and
c^−3 d c^−4 d^2 c d^2 .
Repeat the exercise with the transversals Y = {1, c^−1} and Z = {1, d^−2, d^5}.