I live in an apartment. The dishwasher is a cheap-o model- the drain from the DW goes upwards and hooks into the air/drain thingy that is above the sink; from there there is a thick rubber looking hose that goes from that and hooks into the disposal. From the disposal the plastic pipe slopes( barely) and hooks into the sink drain ( there is a double sink- drain in one and disposal in the other)
I am concerned that the dishwasher drain hose wasn't installed correctly.Also that the disposal doesn't have a p-trap connected to it.
The dishwasher never really drains out all the water- the dishes don't get as clean as I think they should- ?

That "thingy" is called an air gap and by your description, but without a picture of the piping, appears to be connected properly. Many locations, Atlanta for example, do not require an air gap, we use a high loop system. The disposal doesn't have a P-trap because it is connected to the drain trap. You never double trap a drain. It may not work as well as hoped because as you note, is is an inexpensive model. Make sure the bottom screen is cleaned well so the water flows through it.

Brain did a great job of addressing your concerns, let me take a shot at it.

I am concerned that the dishwasher drain hose wasn't installed correctly Your description describes a typical counter top air gap installation (see image. Because they can fail we use a high loop, (see image) in my area.

the disposal doesn't have a p-trap connected to it. Disposals don't have a built in trap. ( here a picture would be helpful) However there should be a trap before the pipe enters the wall. Is it there?

the dishwasher never really drains out all the water It won't! Once them pump stops all the water up to the air gap will drain back into the sump,

the dishes don't get as clean as I think they should Is this the water coming in? Describe what it looks like and what you're concerned about.
Back to you, Tom

I think you have both answered my questions and concerns! Thank you very much! I could put up a photo but this is my first time on this website and One never knows how complicated things can be. Thanks ever so much!
Buscandy

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How do I add pictures and other file types to my post?
A. "Attach" files (including pictures) from your computer. This method puts the attached file below the text of the post.
Click on an Ask or Answer button to ask or answer a question
Compose your question or answer in the window.
Under the window where you composed your question or answer is a section called Additional Options.
Click on the Manage Attachments button: a pop-up dialog will open. NOTE: The Manage Attachments dialog lists the maximum sizes of the various types of files that may be attached.
Click the Browse button to navigate to the image on your computer and double click the file on your computer: this will return you to the dialog window.
Click the Upload button.
Click the link to close the dialog window OR repeat numbers 5 & 6 to attach another image.
B. "Embed" image(s) from the internet. This method inserts an image wherever you wish within the text of your post.
Click the Insert Image icon above the posting window: A pop-up asking for the URL will open.
Enter the URL of the image in the field then click the OK button.