 Then P$Q is logically equivalent to P AND (NOT Q), if the input Q is earlier than P.
The set of OR and $ can represent any logical connectives in any arity except for ones which generates 1 with all inputs 0.Similarly, XOR can be realized with the gate in a diagram above as a bi-root breaking system.The problems of these two root breaking systems is that they heavily depend on the simultaneity of two inputs.