You're mixing a few concepts here so I'll start with a basic explanation about patent protection.
The protection fo a patent is defined by - and only by - the claims. You can have as many claims (more than a certain number cost extra, but in theory) as you want. To get granted they have to fulfill the requirements of novelty and non-obviousness which I will shortly explain. Furthermore, they have to have a common concept, meaning they have to be based on a single invention (unity requirement).
Novelty: the invention may not be known to the public.
Non-Obviousness: the invention has to be non-obvious in light of what's known to the public. This means that a person skilled in the art would not reach the claimed invention by combining known prior art references.
Unity: You can for example have one claim claiming a car with 4 wheels and one claiming it with 5 wheels but not a chair and a candle in the same patent - because the concept behind the claims has to be the same.
When I say that the invention has to be novel and non-obvious, what I mean is, every claim has to be novel and non-obvious. If we are precise, every claim is like a mini-patent united in a bigger patent, every claim has to be novel, non obvious but in exchange defines it's own protection. Someone infringing any claim can be sued by you for patent infringement. There is no necessity of them infringing every claim.
When you say that an invention has to be broken down, probably what you mean are not different claims but the style of drafting claims. A claim consists of different elements. For example, a chemical compound could be produced by combining compounds A, B and C. This would leave you with the claim of "A, B and C combined." Somebody using only A and B or A and C would not infringe your claim.
However, this is not exactly breaking the invention down, it's rather a thing of describing every part of the invention. If your invention is a car, you don't claim "a car", you claim something like "a vehicle with wheels, a motor, a chassy, seats, a wheel" etc. Your second claim might be "wheels, a motor, seats and axes." (Or whatever). To infringe the patent, someone would have to use all elements of a single claim. - This is the protection. Every claim has it's own protection, but the protection is only the combination of the elements.
So in a sense with a little of interpreting, you're right, someone who uses 7 elements of a claim but not the eigth does not infringe your patent. which is why you break the invention down to contain only the necessary elements. Someone using less elements would have made their own invention because they would have found out that one of the elements you deemed necessary was not necessary.