Give a careful proof for a skeptic that the indicated property of a binary structure (S, *) is indeed a structural property. (In Theorem 3.14, we did this for the property, "There is an identity element for *.")

The operation * is associative.

Data from Theorem 3.14.

Theorem: Suppose (S, *) has an identity element e for *· If∅ : S → S' is an isomorphism of (S, *) with (S', *'), then ∅(e) is an identity element for the binary operation *' on S'. 

Proof: Let s' ∈ S'. We must show that∅( e) *' s' = s' *' ∅( e) = s'. Because ∅ is an isomorphism, it is a one-to-one map of S onto S'. In particular, there exists s ∈ S such that ∅(s) = s'. Now e is an identity element for * so that we know that e * s = s * e = s. Because ∅ is a function, we then obtain ∅(e * s) = ∅(s * e) = ∅(s). 

Using Definition 3.7 of an isomorphism, we can rewrite this as ∅(e) *' ∅(s) = ∅(s) *' ∅(e) = q:,(s). Remembering that we choses ∈ S such that ∅(s) = s', we obtain the desired relation ∅(e) *' s' = s' *'∅(e) = s'.