Kitchen Large Electric Appliances - plumbing the drain hose on a new dishwasher install NOT near a sink

This dishwasher is not near the sink so I'm going to plumb the drain directly into the main drain pipe in the basement. I'm confused about what's really necessary. Do I need to install trap and an air gap? Neither?

The helpful person at the local home improvement center set me up with a a 3/4" barb fitting into the main drain line and an air gap. This is a new bosch diswhasher. The outlet hose of the dishwasher appears to be 7/8". The inlet on the air gap is 5/8" and the outlet is 7/8". In other words NOTHING is the same size.

The connection point is the same spot the 1.5" drain pipe from the sink ties into the 3" main I'm tempted to just heat up the 3/4" line a bit to fit over the bosch drain hose so that it's pretty much a direct connection from the dishwasher to the 3" main.

Is this WAY OFF base or reasonably OK? I do intend to run the dishwasher hose up to the height of the cabinet before going back down through the floor. Will that suffice as enough of a trap to keep sewer gas out or do I need to figure out a way to get a full trap in there?

I hope this ramble made some sense. I need to get this install finished but I don't want to rush it and have to do it over again either.

Thanks

As long as you run the hose up as you said you should be ok. That should create a good trap to keep gasses out & prevent dishwasher from siphoning out as it fills. Dave