Set-theoretic view of dependent vs. independent claims

Question

I know that the role of dependent claims has been thoroughly explained in another post in this forum. However, I wanted to clarify something, to see if I am getting this right:

A dependent claim must always narrow down the scope of the associated independent claim. Hence, if someone infringes on a dependent claim, he will surely be infringing on the respective independent claim (it's like saying, if B is a subset of A, and x belongs to B, then it also belongs to A). Therefore, when a court is deciding whether someone is infringing on a patent, wouldn't it be redundant to check the dependent claims? Shouldn't one just examine the broader, independent claims? (I am relatively new to patent law, so I have actually no clue how patent courts decide...).

If dependent claims are mathematically redundant for a court ruling about infringement on a granted patent (I don't know if this is true), then do the dependent claims play a role only during the application process?

Answer 1

A  dependent claim must always narrow down the scope of the associated independent claim.

Yes. That can even be used to determine the scope of the independent claim by analyzing how the dependent claim narrows it.

Hence, if someone infringes on a dependent claim, he will surely be infringing on the respective independent claim (it's like saying, if B is a subset of A, and x belongs to B, then it also belongs to A). Therefore, when a court is deciding whether someone is infringing on a patent, wouldn't it be redundant to check the dependent claims? Shouldn't one just examine the broader, independent claims?

Yes, but. A (popular) defence in patent infringement suits is trying to prove the claims invalid. Therefore, if someone infringes your patent, you'll sue them for every claim they infringe. If they can't prove any of your claims invalid, the dependent claims are redundant. If however it turns out the broad independent claim was invalid, maybe a dependent claim holds.

The dependent claims are therefore fallback positions. They enter the game when the broader claims fall.

If dependent claims are mathematically redundant for a court ruling about infringement on a granted patent (I don't know if this is true), then do the dependent claims play a role only during the application process?

Not only as noted above. It's basically the same role during the application process because they are still fallback positions in case the broader claims are invalid.

Further information on dependent claims can be found in this answer.