It is true that we can do farming without soil.
I'll show you how can we do that but I don't have sufficient sources so I can't show you practically so I'll show you by my sketches and sorry for that. :(
First, take a small net. You can use metal or plastic net. Then take a pipeline which have spray nozzles. Put this pipe below the net and fix this structure.
Now put some seeds on the net. According to your seed, take appropriate mineral nutrients and mix up with water. Spray this mixture via nozzles of the pipe to the seeds.
After few time you can see roots of the plant. Spread this roots to the net and continue spraying water mixture. When plant grow up it needs some support to stay straight so put another net below the 1st net and spread roots on it too. You can also give some support to stem by threads, plastic or wooden stands and so on.
Your plant will grow without soil.
Step 5: Modifications
In this project you can change many things as per your requirements and availability of materials.
1. To make simple water mixture you can take 1 liter water and dissolve powder of calcium nitrate 1 gram, potassium nitrate 0.2 gram, magnesium sulfate 0.2 gram, iron phosphate 100 gram and urea 0.2 gram.
2. In variation, you can use a jar and put water mixture in it and then put a small plant for grow. To give support u can use a CD.
3. You can also use a bucket. Remove bottom of the bucket. At the top of the bucket put a net and seeds and spray water mixture through the bottom of the bucket.
With your imagination make it for you and DON'T forget to give SUN LIGHT to your plant. :)