Learn English – Difference between “with” and “of”

These sentences are a little confusing, because they seem to be very technical. A set containing elements is usually only used when describing things mathematically - a set of prime numbers, for instance. Technical fields will often have specific grammar expectations that may be unique, so it is important to provide context.

In terms of English language usage, "of" means the things that make up the set, while "with" means things that are contained within the set. "Of" is the entire set, while "with" may be part of the set.

For instance, let's say that I have a basket containing puppies.

If I describe it as "a basket of puppies", you could infer that the only thing in the basket is puppies. The basket might contain other objects like a blanket or a chew toy, but as far as the speaker is concerned the only significant objects in the basket are the puppies.

If I describe it as "a basket with puppies", there is a possibility that the basket contains other important things that are being overlooked. For instance, there could also be a sleeping cat.

If I hand my friend a basket with puppies (and also a cat), and she has an allergic reaction to the cat, I can claim to have been truthful. If I tell her it is a basket of puppies and don't mention the cat, she can claim I misled her.