I wrote the following proof to demonstrate the domain of a composition is a subset of the domain of set A, the first set in a composition relation. I hope people can follow this. The instructor and teaching assistant criticized the proof because referencing number of elements is not considered legitimate (see next to last line). The big contradiction, to my way of thinking, is a subset has less than or an equal number of elements compared to the superset. I don't know if anyone can look at this and 'validate' this proof. But I would appreciate any explanation why my logic using the number of elements in the relation violates set theory proof rules....or is it etiquette?

Proof that Dom (S∘R) ⊆ Dom (R)

let a ∈ A, b ∈ B, c ∈ C let R be a relation from A to B let S be a relation from B to C

In order to have(S∘R), the following condition(s) must be true: a ∈ Dom(R) and b ∈ Dom(S) ⇒ a ∈ Dom(S∘R)

By DeMorgan's law, a denial of (a ∈ Dom(R) and b ∈ Dom(S)) would be equivalent to: a ∉ Dom(R) or b ∉ Dom(S)

If a ∉ Dom(R) then a ∉ Dom(S∘R) ["a" would be excluded from both Dom(R) and Dom(S∘R)] If a ∈ Dom(R) and b ∉ Dom(S) then a ∈ Dom(R) and a ∉ Dom(S∘R) ["a" would be included in Dom(R) but be excluded from Dom(S∘R)]

If a ∈ Dom(R) and b ∈ Dom(S) then a ∈ Dom(R) and a ∈ Dom(S∘R) ["a" would be included in Dom(R) and Dom(S∘R)]

The previous statements demonstrate that the number of elements in Dom(S∘R) ≤ number of elements in Dom(R)

Therefore Dom(S∘R) ⊆ Dom(R)