It is easy to prove that a model structure is determined by the following classes of maps (determined = two model structures with the mentioned classes in common are equal).

• cofibrations and weak equivalences
• fibrations and weak equivalences

The second statement follows immediately from the first by duality.

What about the following classes of maps/objects (A short argument would be very helpful)?

1. cofibrations and fibrations
2. cofibrant objects and weak equivalences
3. cofibrant objects and fibrations
4. cofibrant objects and fibrant objects

I think each of these classes determine the structure respectively. For the last one I suppose that one has to use framings but I cannot see how to do it.

It is easy to prove that a model structure is determined by the following classes of maps (determined = two model structures with the mentioned classes in common are equal).

• cofibrations and weak equivalences
• fibrations and weak equivalences

The second statement follows immediately from the first by duality.

What about the following classes of maps/objects (A short argument would be very helpful)?

1. cofibrations and fibrations
2. cofibrant objects and weak equivalences
3. cofibrant objects and fibrations
4. cofibrant objects and fibrant objects

I think each of these classes determine the structure respectively. For the last one I suppose that one has to use framings but I cannot see how to do it.

Edit: Thank you all for the illuminative answers.

1. true
2. ?
3. true
4. false