Do I need a drainage trap?

[Huckabee]

Looking to figure out how I can join these to the drainage that goes to the sewer. I guess I need to get a drainage trap like this , but not sure what else is needed to join the black pipe (shower) and grey pipe (sink and diswasher)

The drainage pipe goes down into an elbow into the sewer pictured left. Well, a little bit of it

 

[Hugh Jaleak]

Direct connection to the drain is permissible, as shown. Ensure traps on appliances have 76mm depth of seal though to help prevent any odour issues.