No, they are not set by price discovery when you have government assistance that corporations and businesses rely on to bridge the gap between what they pay, and what is actually needed to survive.
It has ZERO to do with price discovery, a vague term that you will undoubtedly redefine and wiggle within the parameters later on as your superficial argument gets weaker.
Price discovery needs no government to function. Since you have admitted that you don't know what it means, let me educate you.
Let's say you make hamburgers as a business. Now, you feel that your hamburgers are the best tasting things out there, so you decide to raise the price to $40 a hamburger.
But there's a McDonalds down the street that sells them for $5. There's a Wendy's on the next block that sells them for $5. The gas station on the corner sells them for $4.
How many hamburgers do you think you are going to sell? You need to know this so you can order the meat, the buns, have them ready for people to buy them, etc.
You can't just ask for any old price. You are in competition with other people selling hamburgers. They have found that people in the area are generally willing to pay about $5 for a hamburger. If they don't care too much about the quality, they'll probably just get it at the gas station for $4.
Same with wages. Your 'product' is yourself. You are competition with everyone else that has your skill set. If someone is willing to do your job for $3, why should anyone pay you $15? You can't just demand any old price.
When a government imposes price controls, it distorts the market. The result is inevitable...shortages. In this case, a business that can't make a profit paying people $15 will not be in business long. This just means the business closes or relocates out of the area, leaving you with no job at all. Profit margins at most restaurants is pretty small as it is.
Rent controls are another form of price controls that fail the same way. There is no reason to buy a $1 million apartment complex if you can't ask for a high enough rent to turn a profit. If you are stuck with such a building, you convert it to condos (if the idiot government allows you to) or you tear the building down to put up something else. The end result is no apartments. Suddenly the housing shortage that probably caused the rent controls has become much more acute.
Price discovery is the way of capitalism. Rents are set by what it takes to own, run, and maintain a property, and is in competition with every other apartment complex owner out there. If your rents are too high, you don't get business. If your rents are too low, you are losing money. Government price controls force the rent to be too low.
Minimum wage laws are no different than rent controls. They are price controls. They never work.